If it were me, I'd fly east to the lake and then fly VFR along the
shoreline of Lake Michigan below Chicago's class B airspace. You'd get
a great view of Chicago skyline and could stay within gliding distance
of shore, I believe (never actually have done it, but I'd like to).

Crossing Lake Michigan in a single engine landplane, generally if you
have to ask, don't do it. If everything goes ok, all you will be is
nervous. If you engine quits, you will be in the lake, though I suppose
in July you have a chance, there will be all those boats. Lots of
people have done it though. I crossed it once in my landplane but it
was up by Beaver Island, not so far. I went VFR and got flight
following.. Flight went ok. It was April. I didn't have to ask, though.
It was only slightly nerve wracking. Not as bad on the nerves as
getting ice but worse than moderate turbulence.