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UPDATE: I am now receiving my expected results due to geometrikal's answer. While researching this problem I found a few code examples of Arduino to Floppy Drive. I noticed that they were setting their input pins to HIGH, but I never realized why they were doing this. After fixing my problem, I found this in some Arduino documentation (which made me realize the code examples I had were using an older version of the Arduino IDE):

"Prior to Arduino 1.0.1, it was possible to configure the internal pull-ups in the following manner:"

pinMode(pin, INPUT);           // set pin to input
digitalWrite(pin, HIGH);       // turn on pullup resistors`

In Arduino 1.0.1+ you can do it this way.

pinMode(pin, INPUT_PULLUP);

Here's the code I am running.

//constants
static const int IN = LOW;
static const int OUT = HIGH;
static const int pulseDelayTime = 6;

//pins
int indexPin = 2; //8 on the drive INDEX
int track0Pin = 3; //26 on the drive. TRACK 0
int dirPin = 6; //18 on the drive. DIRECTION
int stepPin = 7; //20 on the drive. STEP
int motorEnableBPin = 9; //16 on the drive. MOTOR ENABLE B

int motorSpinTime = 1000; //in ms


void setup() {

  //initial delay
  delay(3000);

  //setup serial monitor
  Serial.begin(9600);      

  //setup pins.
  pinMode(dirPin, OUTPUT);
  pinMode(stepPin, OUTPUT);
  pinMode(motorEnableBPin, OUTPUT);
  pinMode(indexPin, INPUT);
  pinMode(track0Pin, INPUT);

  //turn the motor off initially
  digitalWrite(motorEnableBPin, HIGH);

  //print state here.
  printState("Setup done.");
  
  //spin the disk some.
  printState("Begin to spin motor");
  spinMotorForThisManyMs(motorSpinTime);
  spinMotorForThisManyMs(motorSpinTime);
  printState("Done spinning motor");

  //step read/write head all the way in.
  stepAllTheWayIn();
  
  //spin the disk some more.
  printState("Begin to spin motor");
  spinMotorForThisManyMs(motorSpinTime);
  spinMotorForThisManyMs(motorSpinTime);
  printState("Done spinning motor");
  
  //step read/write head all the way out.
  stepAllTheWayOut();
  
  //spin the disk even more.
  printState("Begin to spin motor");
  spinMotorForThisManyMs(motorSpinTime);
  spinMotorForThisManyMs(motorSpinTime);
  printState("Done spinning motor");
  
  //never completes.
  waitForIndex();
}

void loop() {
}

//spins the disk motor for a number of ms and prints the state
void spinMotorForThisManyMs(int msToSpin) {
  
  //start spinning
  digitalWrite(motorEnableBPin,LOW);
  
  //delay.. keep printing the state
  int beginningTimeMs = millis();  
  while(millis()-beginningTimeMs < msToSpin) {
    printState("Spinning");    
  }

  //stop spinning
  digitalWrite(motorEnableBPin,HIGH);
}



//step the read/write head all the way to the center
void stepAllTheWayIn() {
  for(int i=0;i<100;i++) {
    printState("Stepping In");
    stepInALittle();
  }
}

//step the read/write head all the way to the outside
void stepAllTheWayOut() {
  for(int i=0;i<100;i++) {
    printState("Stepping Out");
    stepOutALittle();
  }
}

//print the state of the index and track
void printState(const char* charPrint) {
  Serial.print(" Index:");
  Serial.print(digitalRead(indexPin));
  Serial.print(" Track:");
  Serial.print(digitalRead(track0Pin));
  Serial.print(" ");
  Serial.println(charPrint);
}

//move the head towards the outside a little
void stepOutALittle() {
  digitalWrite(dirPin,HIGH);
  stepPulse();
}

//move the head towards the center a little
void stepInALittle() {
  digitalWrite(dirPin,LOW);
  stepPulse();
}

//pulse the step pin
void stepPulse() {
  digitalWrite(stepPin,LOW);
  delay(pulseDelayTime);
  digitalWrite(stepPin,HIGH);
}

//waits for the index to trigger. this never gets completed.
void waitForIndex() {
  
  printState("beginning to wait for index pin to pulse");
  
  //start spinning
  digitalWrite(motorEnableBPin,LOW);
  
  //wait for pulse
  while(digitalRead(indexPin));
  //wait for end of pulse 0
  while(!digitalRead(indexPin));
  
  printState("end of waiting for index pin to pulse");
  
  //stop spinning
  digitalWrite(motorEnableBPin,HIGH);
}

Here is a photo of the IDC Connector and I labelled which wires are connected to the Arduino.

Here's the code I am running.

//constants
static const int IN = LOW;
static const int OUT = HIGH;
static const int pulseDelayTime = 6;

//pins
int indexPin = 2; //8 on the drive INDEX
int track0Pin = 3; //26 on the drive. TRACK 0
int dirPin = 6; //18 on the drive. DIRECTION
int stepPin = 7; //20 on the drive. STEP
int motorEnableBPin = 9; //16 on the drive. MOTOR ENABLE B

unsigned long motorSpinTime = 1000UL; //in ms


void setup() {

  //initial delay
  delay(3000);

  //setup serial monitor
  Serial.begin(9600);      

  //setup pins.
  pinMode(dirPin, OUTPUT);
  pinMode(stepPin, OUTPUT);
  pinMode(motorEnableBPin, OUTPUT);
  pinMode(indexPin, INPUT);
  pinMode(track0Pin, INPUT);

  //turn the motor off initially
  digitalWrite(motorEnableBPin, HIGH);

  //print state here.
  printState("Setup done.");
  
  //spin the disk some.
  printState("Begin to spin motor");
  spinMotorForThisManyMs(motorSpinTime);
  spinMotorForThisManyMs(motorSpinTime);
  printState("Done spinning motor");

  //step read/write head all the way in.
  stepAllTheWayIn();
  
  //spin the disk some more.
  printState("Begin to spin motor");
  spinMotorForThisManyMs(motorSpinTime);
  spinMotorForThisManyMs(motorSpinTime);
  printState("Done spinning motor");
  
  //step read/write head all the way out.
  stepAllTheWayOut();
  
  //spin the disk even more.
  printState("Begin to spin motor");
  spinMotorForThisManyMs(motorSpinTime);
  spinMotorForThisManyMs(motorSpinTime);
  printState("Done spinning motor");
  
  //never completes.
  waitForIndex();
}

void loop() {
}

//spins the disk motor for a number of ms and prints the state
void spinMotorForThisManyMs(unsigned long msToSpin) {

  //start spinning
  digitalWrite(motorEnableBPin,LOW);

  //delay.. keep printing the state
  unsigned long maxTimeMs = millis() + msToSpin;  
  while(millis() < maxTimeMs ) {
    printState("Spinning");    
  }

  //stop spinning
  digitalWrite(motorEnableBPin,HIGH);
}



//step the read/write head all the way to the center
void stepAllTheWayIn() {
  for(int i=0;i<100;i++) {
    printState("Stepping In");
    stepInALittle();
  }
}

//step the read/write head all the way to the outside
void stepAllTheWayOut() {
  for(int i=0;i<100;i++) {
    printState("Stepping Out");
    stepOutALittle();
  }
}

//print the state of the index and track
void printState(const char* charPrint) {
  Serial.print(" Index:");
  Serial.print(digitalRead(indexPin));
  Serial.print(" Track:");
  Serial.print(digitalRead(track0Pin));
  Serial.print(" ");
  Serial.println(charPrint);
}

//move the head towards the outside a little
void stepOutALittle() {
  digitalWrite(dirPin,HIGH);
  stepPulse();
}

//move the head towards the center a little
void stepInALittle() {
  digitalWrite(dirPin,LOW);
  stepPulse();
}

//pulse the step pin
void stepPulse() {
  digitalWrite(stepPin,LOW);
  delay(pulseDelayTime);
  digitalWrite(stepPin,HIGH);
}

//waits for the index to trigger. this never gets completed.
void waitForIndex() {
  
  printState("beginning to wait for index pin to pulse");
  
  //start spinning
  digitalWrite(motorEnableBPin,LOW);
  
  //wait for pulse
  while(digitalRead(indexPin));
  //wait for end of pulse 0
  while(!digitalRead(indexPin));
  
  printState("end of waiting for index pin to pulse");
  
  //stop spinning
  digitalWrite(motorEnableBPin,HIGH);
}