After using a heat gun to remove the biggest part of the paint from the woodwork, we discovered that it was extremely difficult to get it completely clean, especially in the cracks where the various pieces of woodwork fit together. The only option we had was to remove the woodwork. Because the woodwork is so old, is it difficult to remove without damaging it. The easiest was we found to do this was to use a thin putty knife to actually begin prying the woodwork. Once enough space was created, we used a wonder bar to further pry it. The final removal tool is a wooden 1 x 2. Using the 1 x 2 helped to keep the wood from splitting while providing a firm, yet soft, pry bar with a lot of leverage.
Once I got the hang of woodwork removal, I was able to strip the entire room pretty quickly.
This photo shows the woodwork around the windows completely removed.