But my advice would be: Spend one night in Lima and the next morning take the bus to Huaraz, about 8 or 9 hours away. Spend a day or two there, relaxing and making arrangements to trek into one of the valleys, which leaves you about five days (assuming one day to get back to Lima). A day or two walk-in will get you within striking distance of any number of peaks but you'll still not be acclimatized.
From Quebrada Santa CruzIf if condtions are good you could probably run up a 5k peak such as Millisraju or Pucarashta.
From Que Paron, there's Pisco and Paron Grande.
If you had a couple more days and picked your location carefully, you might be able to get up a 6k peak as well if you were very lucky with weather. Both valleys are surrounded by beautiful 6k meter mts.
Of course there are a ton of viables, such as whether you have a partner, want a guide, what level of difficulty you're interested it, etc. But for serious climbing don't waste your time going anywhere besides the Cord. Blanca. If nothing else, the trekking will be great and the peaks you see will make you long to return with more time.
Nine days would be enough to get to Huaraz, do a couple of days of acclimatization around Huaraz, head out to Ishinca base camp and climb Ishinca, and then get back home (I assume the 9 days includes the flights to and from Lima). It would be a tight schedule, but probably doable. There are also other possibilities close to Huaraz, such as Vallunaraju. If you have to spend a day shopping for food, fuel, etc. that would make the schedule a bit tighter. If you have experience with altitude and know you can acclimatize quickly, that would help.