It sounds like the diode bridge is totally FUBARed. Is there any chance at all of reading the part numbers? You can confirm the polarity by tracing through to the +ve and -ve termials of the reservoir cap.
I wouldn't recommend trying to chisel off any heatsinks. If they are epoxy bonded, the IC will probably break first.
However the system is so cheap and nasty (4W @ 10% THD !!!
) that you may decide to simply patch in a cheap 9V 10VA transformer and take a gamble!
Treating it as if the speaker system was actually valuable, you need to make sure your fix never overvoltages it by any significant amount. Take a very close look at the original transformer. If you can identify the terminals of the thermal fuse that's in series with the primary and bypass it you may be able to get it to run long enough to take some measurements (not connected to the amp). Start by measuring the primary DC resistance in case further tests fail destructively. Then power it up on a
dim bulb tester with a 40W or 60W incandescent bulb, with the secondary disconnected. The bulb should not be glowing. Get the exact primary and secondary voltages then soak test it for an hour to reduce the risk of an undetected intermittent shorted turn. Then try it without the dim bulb tester, but with T1A fuses in series with the primary and secondary a 10R 10W resistive load on the secondary to let you get the voltage drop when close to fully loaded. That gives you enough data to pick a compatible replacement, but you could go ahead and replace the bridge rectifier and try it on the amp, so you can measure rail voltages when muted and at full volume. Just don't box it up and put it back into service as without a thermal fuse, the transformer is a fire risk.
Once you have a good idea of the rail voltages you can more easily find a compatible replacement. Add a pair of antiparallel diode in series with the new transformer's secondary if you need to drop its output by about 0.5V or replace the ordinary diodes of the bridge with 3A Schottky diodes if you need to bring the rail voltages up a bit. If the new transformer doesn't have a thermal fuse, leave the primary and secondary fuses mentioned earlier in circuit.
If you cant get the old transformer running well enough to get the unloaded voltage, the only thing you can do is treat that 9.5V on the label as a hard limit for the unloaded voltage of the new transformer. A toridial + adding an overwind so you can boost or buck the secondary voltage a bit may be the best option.
if you were paying a competent tech. labour on the above, you'd already be at several time the new replacement cost of the system so regard it as a learning adventure, and try not to get too far down the rabbit hole buying replacement parts!