I can't imagine spending that much time in Annecy. In my view, there's not that much to see there. Chamonix and the mountains and glaciers around it, are far, far more interesting. A "day trip" to Chamonix is like taking a day trip to see the sun rise. In other words, you're going to miss the best part. I would go to Chamonix on one day, enjoy the ski town vibe and perhaps tour the Alpine Museum to get a little perspective, and then get up early the next day to take the télépherique from Chamonix to Aiguille du Midi and then on to Helbronner. After a generous time viewing these awe-inspiring sites, you can return to Chamonix and take the train to Mer de Glace in the afternoon to walk inside a glacier. That's not something one gets to do every day.
If you have any interest in geology, glaciers (including alpine glacial retreat), or mountains, it's the trip of a lifetime.