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support.hp.com
Support site says PSU is "internal ATX", so yes, try a fresh PSU on it. Even a $28 Raidmax 400W special might be more than enough for that rig.
If that doesn't work, pull the HDD for recovery and privacy purposes and junk the rest. (CPU and RAM might have some remaining value on ebay).
Edit: Also, consider the video output, sometimes the chipset iGPU in those older boards could overheat if the chipset heatsink got covered in dust and the case fan died or something, and then it cooks itself. Consider dropping in something like a GT610/GT620/GT710/GT1030 if you happen to have any of them around. It's
probably not that, as if the video output was hosed, then either you would get a POST beep error, or a normal POST "beep", just with no video. If the power button LED is not lighting up, then it's probably not POSTing successfully, just with no video output, and does point more towards a PSU or other serious component failure.