Well, you'll have to go well beyond the keyboard there.
It's been a while since I had to tear down a T-series laptop, so you'll have to bear with me...
It's a good idea to do a web-search with your model number and the term disassemble. There may be some tutorials out there.
I can't remember if the headphone jacks are integrated on the mainboard or if they are separate. If separate, then you should be able to get another module for cheap on eBay. The speakers should be separate and the same thing applies.
The speakers are likely easier to get to and probably cheaper as well, so I'd try that first.
Does the computer beep when you turn it on? If so, then the speakers are not the problem...