I would opt for the southern half – it is incredibly beautiful and much more remote than the northern half is. If your wife is hiking north to south, you could join her at Vermillion Valley Resort or Muir Trail Ranch. If she is hiking south to north, you could just start the trail with her and get off at either Vermillion or Muir Trail Ranch. Unfortunately, neither place is particularly easy to get to. Depending on your pace and the logistics (it might take some time to get to Vermillion or Muir Trail Ranch), you might find a shorter section more manageable with your time constraints.
If logistics prove too difficult, you might consider a middle section. There are a number of places where you could join your wife by hiking in to the trail from the east side. Mammoth/Red’s Meadow provides several options. Further down the trail you have Bishop Pass (which drops you into LeConte canyon, in between Muir and Mather passes), Taboose Pass (drops you a few miles below Pinchot Pass), and Kearsarge Pass (drops you in between Glen and Forrester passes).
Hope this helps. Whichever section you decide to do, I’m sure it will be a wonderful trip!
Kanga