Cardboard will not work in the long term. It will suck humidity out of the cage and become wet over time. Aluminum tape or aluminum foil is better. Duct tape is even better than cardboard.
OP: The main problem is the heat light. If you turn it off you should observe your humidity shooting up. If your normal room temperature is above 70*F, I would highly suggest using only UTH belly heat instead of lights.
Other things that help humidity are:
A larger diameter water bowl
Covering the screen top with 2/3 aluminum foil or foil tape
Using a good moisture holding substrate like cypress mulch
No offense but have you ever used tin foil before? if you have ever taken anything out of the oven that has tinfoil over it you would know that you can touch it with your bare hands after a few seconds.but yeah i went with cardboard cuz i didnt think of the tin foil... But i would have an issue with that getting hot since it would be around my lights.. And my digi wire runs on the top of the cage