Don't get disheartened. My wife is only 5'1". Or is that 5"1'. Whatever.
Places I take her... First overnight walk I took her on was Frenchmans Cap - made a 7 day circuit out of it including swimming across the Franklin river.
Not the recommendation for anyone on their first overnight walk in Tasmania, but I figured if it was good enough for me so many years ago...
Now she is a BW junkie. Any chance at a walk, gone!
Western Arthurs to L.Cygnus, overland track, she even did the south coast track without me due to a knee op. (but with a couple of other friends).
She does Hansons Peak and down to L.Rodway by herself as a day walk on good days (with a pack that contains camera gear and everything she could need if something went pear shaped).
What I'm trying to saying is the height, or lack thereof, shouldn't be too much of a problem - keep on with the fitness thing.
Yes you do need a certain level of fitness to do many walks in Tassie if you want to enjoy them rather than treat them as a slog.
But the rewards will be worth it - just check through the gallery!
In the mean time, continue with the less demanding walks as you are - perhaps some more suggestions of walks for you will emerge.
Have you checked out Narawntapu? Day walks, no huts, but a lot of it is flat, or at least less demanding than Hansons Peak.
I agree Lady Lake hut is nice, a bit of an uphill section to get to it but worth it and a very comfortable hut, search this forum.
Here's a look of the terrain...