Good luck Turnstone.
I agree that core strength training is a good idea. Yes, legs, hips, core for strength. Lunges and squats especially.
Also for building stamina, cross training that is non-weight-bearing, e.g. swimming or cycling, will complement the walking and give you plenty of variety.
F 5 ft 3. HW: 196 lbs. Primal SW (May 2011): 182 lbs (42% BF)... W June '12: 160 lbs (29% BF) (UK size 12, US size 8). GW: ~24% BF - have ditched the scales til I fit into a pair of UK size 10 bootcut jeans. Currently aligning towards 'The Perfect Health Diet' having swapped some fat for potatoes.