After flipping it to get dT/dt, how would I solve the integral and rearrange for T? Help appreciated.
IF the problem was $\displaystyle \frac{dt}{dT}= -K(T- 5)+ H$ then write it as $\displaystyle dt= (-K(T- 5)+ H)dT$ and integrate both sides.
IF it was $\displaystyle \frac{dT}{dt}= -K(T- 5)+ H$ then write it as $\displaystyle \frac{dT}{(T- 5- H/K}= -Kdt$ and integrate both sides.
(To integrate on the left, let u= T- 5- H/K.) T will be an exponential of t.