I count 12 full days and that may be just enough time for Lima, Cusco/Sacred Valley/MP and Lake Titicaca.
The 2 day Inca Trail is expensive for what is really only one day of hiking. The second day is a visit of MP which you can easily plan on your own (you would just buy train tickets and MP tickets). But maybe you have your heart set on it.
For the 4 day Inca trail you would have to acclimatize in Cusco for at least 2 nights before doing the trek but I don't think that is necessary for the 2 day Inca trail. You could go to Ollantaytambo on arrival in Cusco (Ollantaytambo is lower in elevation than Cusco but still as high as you will be on your trek). If you arrive in Cusco early you could sightsee on the way (Chinchero, Moray and the Maras salineras). Or just go to Ollantaytambo, explore the town, see the ruins, hike up to the granaries and visit Chinchero, Moray and the Maras salineras the next day. Then go back to Cusco and explore Cusco. Do the two day trek and visit MP. Then another day in Cusco for Pisac and the ruins north of Cusco (don't miss Sacsayhuaman). Then take a bus to Puno. Do a 2 day-1 night Lake Titicaca tour with an overnight on one of the islands. Fly to Lima and spend two days there.
You could also do Cusco first (a day for Cusco and a day for Pisac and the ruins north of Cusco), then the trek and then Chinchero, Moray, the Maras salineras and Ollantaytambo. In that case just take it easy in Cusco those first two days. Being fit doesn't mean you won't feel the altitude. You most likely will. Just don't overdo it. And if you do any hikes/walks take it slow and steady.
Consider hiking up one of the mountains at MP. Huayna Picchu or MP Montana. We chose the latter and loved it.