ORD to the Upper Peninsula of Michigan when one is tired and very jet lagged (and with an infant) would be very tough in the current winter weather, although the interestates should be clear of snow and ice. My concern would be the part of the journey on secondary roads in northern Wisconsin and into northern Michigan. It would be worse if the OP isn't an experienced winter driver.

Have you asked UA whether they would be willing to try to reroute you on DL, through MSP or maybe DTW? However, they're certainly not obligated to do this for weather related IROPs.

If you're confirmed on flights in two days, can you just take a hotel room st your own expense and keep it to avoid checking out and returning to the airport every day? You could then ask UA later to please either reimburse you their contracted hotel rate or to give you some miles as a customer service gesture. Alternatively, what would it cost to purchase two tickets on DL if seats are available? [Note that if you rebook yourself, UA will not pay, but you would get a fare refund for the ORD-Houghton segment (likely to be a very small amount assuming that you have a through fare) and you might get some customer service gesture miles too as a UA 1K, more so if that TATL ticket was purchased in business class.]