You should've titled this "Mouse Problem"...
Actually, the mouse is connected to Denise. Try re-seating her and see if that helps. If not, then check the 74HCT157 (U15) that buffers the signals from the mouse.
Amiga mouse basics:
The Amiga mouse works by pulsing the movements in what's known as "quadrature mode". Essentially, there are four lines (one pair for each axis of movement) that are monitored. If Denise sees pulses coming in on any of them, it compares them to each other. If the pulses on the data line precede the quadrature (Xd before Xq), then it knows the mouse is moving down. If the pulses are lagging (Xd after Xq), then it's going up. Ditto for the Y axis (Yd <> Yq).
Once we know that, we can do some quick diagnosis:
Power is good, since the mouse generates the pulses using IR/LEDs, and if the mouse is sending data, even in only one direction, then its got power.
There's a possibility that the "Q" signals aren't being sent, but the data is (i.e., Q is always lagging D). This requires a logic probe (about $20 at Radio Shack) to check. Best bet is to check at inputs and outputs of the 74HCT157 (U15). This will verify both the mouse and chip.
After that, it's all Denise. If power and signals check good, then swap out Denise (8362/U4).
The only other alternative if none of these work are the little ferrite EMI filters along the back of the port. If these open up (or short to ground), then this type of behavior could happen.
banzai