Not really a display forum topic, unless you have doubts whether the monitor itself works. Since it is the display forum, has it been tried with another system? Does the manufacturer logo usually come up first and does it now? Does an LED power on indicator light up consistent with it staying on? It is not uncommon to have monitors fail, and not end up in a working mode, but usually the LED power indicator shows something different.
Otherwise, how long ago did the system last work? Has it been moved since then, so perhaps it's worthwhile to try reseating the memory?
What evidence did you have that the battery was nearly flat? If you knew that, then why wouldn't you have replaced it already? Have you tried clearing CMOS? I realize some others have mentioned this, but it bears repeating.
How old is the system? Another possibility is that sometimes capacitors just shelf rot, so examine both the motherboard and interior of the PSU for vented capacitors.