Check and make sure all your connections are correct and tight. Remove and re-seat the memory modules and any add in cards. If you have a motherboard manual for the machine I would recommend clearing CMOS.
If you perform the above mentioned items and the machine still has problems there is a high percentage of probability that a motherboard component or components were damaged when the PSU failed.
The only way to prove this is by replacing board components such as memory, add in cards, etc. with known good components.