If you are lucky, it could be something with the power supply. Start by measuring the voltage rails (DC voltage and ripple). There may be instructions for the manual for this, otherwise you will just have to reverse-engineer it. For example, if you see an 7805, the output should be very likely about 5V DC. If you find that the output is much lower, there is likely something wrong. Many large value electrolytic (aluminum and tantalum) capacitors will often be across a voltage rail, so could be a good place to measure voltage. There will be most likely be some 3.3 or 5 V rails for the digital logic, and some analog rails (possibly symmetric, e.g. +6 V and -6 V). Since it is not powering on at all, if it is a power supply issue, there are probably rails that are way off.
Try to find the most recent power meter that still had schematics published. The HP 437A/438A is much older, but still had full schematics in the user manual. Not sure about more modern models. The principle of operation and possibly even the circuit of the probe interface and reference output may very well be very similar (since they could use the same sensors). The digital circuit is probably all different.