I believe you have a typo in this section with the subscripts:

If the sum of the two reports θJack + θJill is at least $300, the room will be painted. Jack’s payment will be pJack = 300 − θJill if the room is painted and zero otherwise. Similarly, Jill’s payment will be pJill = 300 − θJill when the room is painted and zero otherwise.

If Jill reports between $150 and $180, the room is painted and Jack gets a payoff of 120 − (300 − θJill) = θJill − 180 < 0, less than his payoff from honesty where the room isn’t painted.

When the room is painted, Jack and Jill have a $300 bill, but the mechanism only collects 300 − θJack + 300 − θJill < 300.

If pJack = 300 − θJill and pJill = 300 - θJill, then the mechanism should be collecting 600 - 2*θJill?

So it turns out that the reason none of the solutions I tried in the last article worked is because it's impossible for solutions to exist that satisfy the implicit restrictions I was putting on solutions. That's interesting. It's too bad that the second-best solution is so relatively crappy. Realistically, it doesn't seem to be any better than just negotiating with someone in the more traditional manner, and accepting that you may not end up revealing your true preferences.