Welcome to the forum. Not only is your trek from Brookings, SD to New Haven, CT doable in 23 days, but it can be an awesome "scenic adventure". Our small group of four is starting from the Oregon coast this summer and taking a very indirect and adventurous route to the Maine coast. We will be riding through the Black Hills and Badlands before taking Hwy. 14 across SD. We will be rolling through Brookings around the third week of July.
Meanwhile, back to your question: To start, create a bike-suitable route straight across MN and WI to Manitowoc, WI and take the ferry across Lake MI as outlined on Adventure Cycling's (ACA) "North Lakes" maps section 2. Then connect with North Lakes map section 3 at Wolf Lake, MI and follow section 3 to Monroeville, IN. At this point connect with the ACA "Northern Tier" map section 9 and follow this route east to Orchard Park, NY. From there you'll want to map out a beautiful route across NY through the Finger Lakes region and the Catskills, across the southern Berkshire Mts. into CT and you're home free. This would be a great route and very doable in 23 days.
To summarize: you should purchase the ACA maps "North Lakes" sections 2 and 3, and "Northern Tier" section 9. These maps not only map out the route, but provide locations and phone numbers for lodging, camping, groceries, points of interest, all along the route. So this part of the trip is all done for you. The rest of the route will be up to your creativity. The web is your friend.
I hope you decide to do it. It should be an awesome adventure.