So what causes this death of our beloved ihomes? Well in most cases its the actual connection between dock and device. Over time the pins that make the connection wear down and we're left with an iPaperweight.
Step 1: What to Get
1. a broken ihome
2. toenail clippers
3. a cutting utensil
4. a broken sync cable that you don't mind sacrificing or a small needle (edit: or a staple)
Step 2: Jump In
Note: the next few steps can be skipped if you have a needle small enough to fit in between the grooves of the ihome dock connector (the smallest needle I had wasn't even close but if you're a nurse/doctor maybe you'll have better luck than me).
edit: I recently discovered that you can complete this next step using a staple if you use the toenail clippers to trim it to a thinner profile.
The first thing you need to do is pull out your broken sync cable (not your ihome!!). Use the knife and carefully disassemble the top half of the connector. I found that breaking the plastic was the easiest starting point. There are two clips that hold the plastic on on each end of the sync cable. After you have the plastic off you should have access to the wires and the pins.
After you have access to the pins pull one out by the wire at the end and clip the wire.
The pins end should bulge where it would normally make the connection. Take the toenail clippers and cut the bulge off as to make the pin as skinny and sharp as possible. Now take the modified pin and go to your ihome. Carefully moving from pin to pin on the ihome slide the the modified pin behind the ihome's pins to so that they are pushed away from the groove that they rest in. They will pop back into the grooves when you remove your pin but you should notice a little gap behind the pins after you're done.
After going through all the pins plug your ihome back in and put your iDevice onto it. If it works great, otherwise go back and insert your modified pin a few more times.
Also if you're trying to get a certain functionality to work the following pin diagram may be useful so you can focus on the non-working pins: http://pinouts.ru/PortableDevices/ipod_pinout.shtml
That's it enjoy your once again functioning Ihome!