Why are the batteries not lasting more than a couple days?

All of a sudden, my calculator stopped working for more than a couple of days after I put in new batteries; even when I only use the calculator for a couple minutes on one or two days! I tried replacing the backup battery, but this doesn't seem to have done anything. I'm not sure if there's a short in the power system or what the problem is. At one point, I had left some old batteries in there for a long time, and one of them looked like it was starting to leak some chemicals out. I wiped them off and cleaned out the battery compartment as best I could, but since then the new batteries just haven't worked for very long. What could be causing this, and what can I do about it? I'd rather have it fixed for cheap, or fix it myself if possible with my skill range and tools, than purchase a new one right now.