Hi Karyn and welcome!
Your question has got me thinking! Usually people are keen to walk into Santiago. You've got plenty of time to do more than the last 100kms, as that stretch is just 5 days or so. It does get busy on that stretch and you see things that you might not do earlier on the route (more horses, bus tours etc), but there are some lovely places there.
Your time is quite limited so you want to find a starting point that is fairly easily accessible from Madrid. If you wanted to do the 'middle' section that might mean starting somewhere like Burgos. That section would take you across the Meseta, perhaps to Astorga or Ponferrada depending on how far you walk each day. Some people don't like the Meseta, as they think it's dull and flat. Others absolutely love it, find it really spiritual and the essence of the Camino. Some Spanish pilgrims asked my walking partner this year what had been her favourite part of the Camino, and when she said the Meseta they thought she had gone mad. It's a matter of taste!
Another option would be starting in somewhere like Astorga and walking to Santiago. You'd have the beautiful mountain section between there and Ponferrada (including the Cross of Iron) and the classic climb up to O Cebreiro as you enter Galicia. This would be my suggestion given the time you have. Leon would be an easier starting point to get to, but might be just too far to walk to Santiago depending how far you can walk each day. Even if you're not too worried about getting a Compostela it might be frustrating to get so close!
Personally my favourite section is near the start in Navarra and La Rioja, but that would leave you a long way from Santiago. Maybe that's somewhere to go back to once you've got the Camino bug!
Buen Camino, whatever you decide to do!