I've just gone through this with an old synth. Have you checked the PCB for damage to any traces? If the battery has been leaking for a while this can easily happen. Boards can quite often be repaired.
For the battery replacement, the Li types will generally last very well, and won't leak. Also, there are Li batteries made with solder leads, which can be convenient if you don't want to install a socket. Be careful insulating any conductors which may short. You do have to be careful with the diode too. If you're not confident with electronics, you may be better off just using a battery equivalent to your original. All these batteries are not too expensive anyway. Usually about $10 to $20.
I would think using multiple AA or AAA batteries would cost more, and be more difficult to fit.
-Oli
