The only real way to test a PSU without equipment is to connect it to another pc and see if it turns it on. I assume there are no lights or fans coming on at all ??
Power supplies are easy enough to change and are not that expensive.
IMO if you've got no one local who can help, order a new power supply and fit it. If it dont work its most likely motherboard if nothing is coming on at all.
Just to throw a spanner in the works, I have had one PC not turn on and it was the on/off switch. But if your pc died in use my money is on power supply.
If the new power supply does not work your gonna have to think hard about it to be honest.
Changing a motherboard is not the easiest thing to do at the best of times, you've got to find a compatible board thats the right size, will fit your case (CPU & fan sometimes clash with case) accepts your processor and memory (I.e. Intel, Amd and socket type) and if you get it done at a shop there going to charge a nice hourly rate.
The cost of a nice new Dell Pc which is a lot quicker that what you have now might end up costing you £100 more than getting yours fixed. All its all brand new and warranted.
If you get yours fixed and the harddrive goes or the cpu its more money to get it fixed and its still not as quick as a new one.
In short cos I`ve waffled on.Buy a new power supply - If it dont work buy a new Dell Pc. If it does let me know and we`ll sort your memory out.
Dells cheapest PC will be more than good enough for what you want but let me know if you want any help choosing one.
Oh and no I dont work for Dell
Imo there the best on the market atm then HP.
Is this what your power supply looks like ?