Updated some values, most importantly, the k-factor. I had assumed that it was in the correct units already, but when converting it to 0.046 W/(m^2*K) from 0.26 BTU/(h*ft^2*F), I got the following plot. The time constant is still a bit larger than what we'd expect, but it's much better with these adjustments.

For our next steps, I will measure the time constant of the heater without any insulation and then decide how many layers of it we will need. I'll need to construct and calibrate a temperature sensor like the ones I've made before and use it to record the values more accurately.