i had a similar problem w/my 1 1/2 year old t3. starting on friday, it wouldn't turn on at all or even make the chirp sound when you put it in the cradle. my online research found nothing i haven't tried before except for instructions on how to take it apart.
since i had nothing to lose, i was able to open the thing up today (a little tricky) and discovered that when the case was open, it would turn on. however, if i closed everything up again, it was gone and the back was hot. figured there was a short somewhere, so i opened it up again, placed a sticky note...yes that's right...a cut-to-size post-it sticky note between the motherboard and the back of the screen, closed it back up and EUREKA, it works!
i've heard of using baling wire & duct tape, but i can now add post-its to the list!
you could try googling for instructions to open your t5 up and see if you get anywhere with it. at the very least, have fun disecting it.