Dude I feel your pain. It will get better.

We have two kids and although one wasn't as bad as than the other, neither were great sleepers when they were really young. I'm not sure this is the best method but our wife and I had designated nights where one was responsible for getting up with the baby and the other could sleep. More than once I slept with earplugs in the basement so I could get a full night of sleep. She would sometimes do the same on her nights to sleep.

I think you need to come up with an arrangement with your wife where you can both can whatever it is you need. You might do this kind of thing already. You watch the baby for a couple of hours so she can go to the health club, have coffee with a friend, etc. She watches the baby for a couple of hours so you can ride.

Riding every day just might not be practical at this point in your life, but like I said, it gets better. You could perhaps drive part of the way so riding only adds 30 minutes to your commute instead of an hour but for now I would try to get your exercise in a way that doesn't limit your sleep.

This last piece of advice doesn't have much to do with cycling but I strongly recommend that you and your wife make sure to take care of each other and your marriage as well as your new baby. Take advantage of relatives or anyone else (within reason) willing to babysit your daughter for a few hours or overnight so the two of you can have some time together and your baby gets used to being taken care of by other folks.