Time of year Indeed! February is in the rainy season. Not that it's pouring with rain all the time, it's just that there is a fair amount of precipitation, and if it's dry, there is a good chance that it's cloudy so that any mountain tops that may be around are invisible. All in all, in the wet season virtually nobody climbs in any of the glaciated ranges, and there are much less people on the hiking trails. The Inca trail is in fact closed in February.
Nevertheless, I fully understand that you want to see Machu Picchu, it is indeed a remarkable place. I visited in June, and enjoyed a sunny day, the sort that you need to take those pictures that appear in the travel brochures. In February, you're at the mercy of the weather. You may well be lucky and have a sunny day, but if it's overcast, I bet that it will be less scenic, and if it's raining, well, then I wouldn't even want to go there.
VIsiting Machu Picchu means you'll be flying to Cuzco, which in itself is an interesting city, with lots of history. And not all that far away are several mountain ranges, notably the Cordillera Vilcanota, my favorite range in the south. You could hike the Ausangate circuit - but in February you'll get wet. Combining Machu Picchu with the Ausangate circuit would take up most if not all of your 10 days, but perhaps you can still squeeze in a visit to Lake Titicaca.
If you're looking for bigger peaks to climb, the volcanoes around Arequipa can be ascended all year round, in principle, though the success rate is much lower in February than a few months later, because even if it's not raining or snowing high on the mountain, ascending in the clouds does not exactly provide the motivation to keep going if you're already struggling with the thin air. Considering that Misti is 5822 m high, you need to be acclimatized before having a go at it. Starting your trip at Cuzco, then a visit to Machu Picchu followed by a couple of days at Lake Titicaca helps, though probably you'll be struggling at best, and your headache may force you to descend.
If you're in Arequipa but the weather isn't right for Misti, or you don't feel acclimatized enough for it, I recommend a trip to Colca Canyon instead, including a couple of nights in the area, so that you have enough time to hike down to the canyon floor.
With more time I would also recommend visiting the Cordillera Blanca, but with just 10 days it seems a bit silly to spend several days traveling in Peru on buses and planes, which rules out any itinerary with both Machu Picchu and the Cordillera Blanca.
Saludos, Rob
P.S. Assuming you haven't already read them, I posted a
series about Peru that you may find useful. It starts around Arequipa and moves on to Cuzco. Part 4 includes a visit to Machu Picchu. After that, the story continues in the Cordillera Blanca, so you can skip that.