You don't want to stay in Whittier because there is nothing to do. Go to Anchorage on the train or a limo. In Anchorage, we stayed in a wonderful B&B. It's close to all the downtown things to see but it's quiet and lovely. It is the Sleeping Lady Bed & Breakfast. We had a large, unfussy room with a view of Cook Inlet. The downstairs area of the B&B has a comfortable living room and indoor breakfast area with lots of windows. There is a large outdoor patio for eating and lounging. Everything is quite new and clean. I don't recall the price but it was pretty reasonable for Anchorage.
We stopped by the Captain Cook Hotel and it was dark and filled with people. We much preferred the quiet and airy feel of the B&B. I'm sorry I don't have the phone number or web site but you can find it on Google.