We have that
where is the temperature of the body (in this case the thermometer), is the temperature of the environment, and is the "time constant," which we don't know yet.
We know that and that at , , and at , .
Two equations and two unknowns. We want and don't care about the time constant, so what I recommend you do is, instead of trying to solve for directly, solve the top equation for . Then note that and use that for your substitution. You'll be left with an equation for .
Sorry, I gotta run.