Hi, from what I've read so far on this site about measuring temperature with a thermofilter, everybody seems to be recommending the T thermocouple. The problem that I have here is that the thermometer that I have, the extech TM100, only offers K and J thermocouples. Would I need to buy a new thermometer that offers a T thermocouple or may one of these two do the job for espresso temperature reading? Thanks!

If you do want to use a type T thermocouple you will need to have a reader that can be set for type T.
Type K is just fine in this application.

Type T on paper is better, but unless you plan on spending $$$ on a very high level reader and having everything calibrated you are going to get more of an error from your hand held reader than you will from the thermocouple wire itself.