CIA chips were always a common failure on Amigas. When one goes bad, floppies could fail or the whole machine might not boot. After all the common fixes like... power supply, memory, battery corrosion, expansion hardware, resetting connectors and chips in sockets, have all been eliminated as the problem, I would recommend the following trouble shooting to some one with very good solder skills. (U7,U8, U28 and U29 are surface mount)
Replace U28 and U29 (They're cheap). If those chips were to fail, they could keep the floppy from working or the machine from booting by tying up CIA U8. If no change, then it's time to remove U8. With U8 removed, power up and see if the symptoms change. If so U8 is probably bad. If no change, replace U8 and remove U7. Again look for any change in the symptoms. If it looks like one of your CIAs are bad, I don't know that there is a source for replacments. But if you have two machines, you might be able to come up with enough good chips to get one running again.
FYI, carefully using a basic heatgun works well for doing surface mount parts.
Picture of CIA on A1200/A4000Plaz