There's a repair manual describing each internal tests (well, except for the ones introduced in later firmware) and what they do, and also describes some basic checks like supply voltages. If not done yet, it could be a start.
A reversed schematics is also available.
On the 10V range the input resistance should be >10G, you should see changing numbers there. When you connect your meter, the current flowing creates a voltage drop on the input resistance of the meter. If the meter's input resistance is 10M, you have a 44nA current flowing there, which is way too much (should be in the range of 10s of pA), though your meter can have its own bias current added.
There are several topics here on the forum about fixing K2000/K2015s with excessive input bias current, I'd recommend to check them out.