Com_Cool is right on the money. You haffta hit Target Field for a Twins game while here, and at the game make sure you have a Murray's steak sandwich. You will NOT be disappointed.
I hate to disagree with the good Dr. but I'm not a fan of the Target Field Murray's steak sandwich. It is different from the steak sandwich at the restaurant 2 blocks away, but the difference in quality is miles different...if it wouldn't kill me, I would have the Murray's steak sandwich slathered in bacon and cheese every.single.day. And that's after gorging myself on the crispy garlic toast.
Guthrie Theater, Walker Art Center (adjacent) and Sculpture Garden (also adjacent) are good cultural activities.
The Guthrie Theater is no longer attached to the Walker or the Sculpture Garden. It closed in 2006 (at that location) and reopened on the 2nd St. a few miles away, near the Mississippi river. It's too early in the season to have a good calendar of events for the week you are here. However, the Walker and the Sculpture Garden are stops you should make, especially the iconic shot that you should take, or have someone take of you and the princess at the sculpture garden:
<img src="http://kyleandmara.com/wp-content/uploads/2013/11/minneapolis-spoon.jpg" width="25%" height="25%">
There is no dearth of "high end" places to eat, however, there is a great variety, and some of those might not be high end. For example, I know people who go crazy over the lobster roll at Smack Shack, which is fantastic, but not "fancy" or "high end". It would be instructive to learn more about what you're looking for, and I would encourage you to seek out some local things like the previously mentioned Sea Salt, which is right next to the Minnehaha Falls:
It's a lovely park and the falls meander into the Mississippi river as well.
The Twin Cities is a large metro area with a lot to offer, and as posters here have mentioned, if you have the desire to see more of the state, heading a few hours north to Duluth or beyond would be worth your time. You could just as easily head down to Red Wing (stay at the St. James), drive down highway 61, following the river and probably see more bald eagles than you can count.
I'm not trying to tell you what to do, but there are many diverse options here, from truly high end places to grittier but no less enjoyable options in Northeast, Uptown, many areas in St. Paul etc. etc. etc.
This is Minnesota after all, so we're all happy to help, and we'll help you find any place.