"Lock them in a room" and other "solutions" to Illinois' governmental problems

This could be done legally but probably not under the current IL Constitution. You could adopt the papal conclave model and evict them from office if they do not attend rather than forcibly detaining them. There would have to be reasonable exceptions for emergencies. It would likely force a resolution but probably not a great one.

While we are on the subject of pipe dream "solutions", I will add a couple ideas that I think would be worth discussing. First, we should move to a unicameral legislature. There is little benefit to a 2 chamber model. Second, we should set the size of the legislature to roughly 1 per 60,000 residents (215 vs the 177 we have now). It is easier to know your representative and harder for outsiders to "buy" an election with smaller districts. Third, terms should be lengthened (10 years?) to end this constant game of taking tough votes after various election-related dates. This could be coupled with a single term limit and/or recall provisions.