Simulate the Home button
To navigate without the home button, you will want to enable AssistiveTouch. This can be done by going to Settings > General > Accessibility, and scroll until you see AssistiveTouch*. You will want to turn that on.
*AssistiveTouch requires iOS 5 for the "Accessibility" menu.
You can drag this anywhere along the bottom or sides of the screen.
Now how do you navigate? By tapping on the icon pictured above, you will be presented with a few buttons and gestures that must normally be physically performed on the device. You will get options for the home button, the device itself, and multitouch gestures.
By tapping on the icon for the home button, this simulates pressing the actual home button. For example, if you wanted to double tap the home button to lock the orientation or view your recently opened apps, you could double tap on the icon on your screen that simulates the home button.
To mimic the power button, tap the lock screen icon. For example, if you wanted to power off the device, hold the "Lock Screen" icon until the red "Slide to Power Off" slider appears.
To mimic the volume buttons, tap on the Volume Up or Volume Down buttons on the screen. You will also have a button to mute or un-mute the device.
You can also tap the Back (left) arrow to return to the previous menu.
Replacing the Device Under Warranty
If you have the option of getting the button replaced (by getting the device replaced), that would probably be the best route. It would be a lot easier to use a functional button than to use AssistiveTouch. If your device was purchased less than a year ago, or you purchased AppleCare and the device was purchased less than TWO years ago, it is under warranty. The warranty does not apply if the device was dropped or suffered from any other accidental damage from the user. If the device is not in warranty, you may want to see...
If you have an iPod:
If you have an iPhone:
|iPhone 2G (original), iPhone 3G, iPhone 3GS, iPhone 4||$149|
|iPhone 5, iPhone 5S, iPhone 5C||$269|
|iPhone 6 Plus||$329|
If you have an iPad:
|iPad mini (1st generation)||$199|
|iPad (original), iPad 2, iPad mini 2||$249|
|iPad (3rd or 4th generation), iPad Air, iPad mini 3||$299|
|iPad Air 2||$379|
Note that they will not replace button, but rather the device itself. Check your warranty here:
The above link asks for your serial number, which can be found on the back of some devices. If your device does not have the serial number printed on the back, you can go to Settings > General > About > Serial Number to find it. If your device is in warranty, the replacement will be free.