Welcome to FT. When are you going? That will make a huge difference. My guess is that the trip is coming up soon since you mentioned the national park shutdown.
Actually the Kenai PENINSULA is not out of the question. The national parks are closed but lots to do/see besides that. You really don't want to be spending 6+ hours driving to Seward round trip with such a short time. The main reason to go to Whittier is to do a Prince William Sound glacier tour but those are no longer operating (ended Sep 30).
I would get a hotel for your first night, then get a start in the morning along Turnagain Arm toward Girdwood. Lots of stops you can make. Not everything will be open, but here is a link to some great activities:
http://www.tripadvisor.com/ShowTopic....html#47298453
I would stop at Potter Marsh (look for wildlife), Girdwood (tram if good weather, maybe the Winner Creek Trail and hand tram best hiked from the Crow Creek Mine side of the trail), Wildlife Conservation Center, then maybe driving south along the Seward Highway to the Hope Junction. Drive into Hope. Maybe have lunch here. Great views all along Turnagain Arm from Anchorage to the Wildlife Conservation Center then again from Hope.
If the weather is nice, you can consider some activities in Anchorage too - bicycling on the Tony Knowles Coastal Trail or driving up to Glen Alps or going to Lake Hood to watch floatplanes take off/land (great activity after you check in but before you return the rental car).
But I would spend most of my day along Turnagain Arm as mentioned above.