This is a progressive skill challenge system, as PCs get better at singing the complexity rises. They also gain levels in the instrument which they are training with, getting a roll modifier with each level. This speaks to the PC getting better at singing but also learning harder and more complex songs. PCs rise in instrument levels as they get better, this level can be used as a reaction modifier (1d20 + instrument level + CHA modifier) based on the audiences expectation (DC should take into consideration the difficulty of the piece or orchestration and the audience's expectation.) This level can also be used as a modifier to learn the next level, making it that much easier to pass individual skill checks. The instrument level can also obviously be used for when you're playing your instrument for any other reason, assuming how well you're playing is a factor in what you're doing.
The stats used are DEX and Insight alternately. DEX because of the manual dexterity involved in learning how to play an instrument and Insight for judging how good you sound.
I'd recommend a fairly high DC, a level 2 PC of mine had a DC of 18 and a complexity of 1 in order to learn the first level of instrumentation. This is because picking up an instrument you've never played before and figuring out the basics is a lot harder than practicing and building on what you already know. Her challenge complexity will be increased every two levels she increases. So basically the complexity is (instrument level/2=complexity level for the skill challenge.) This assumes your PC starts at instrument level 0, which she should.
If you're learning with a teacher, they would make a saving throw against how awful you sound, and if they pass they give you a +2 mod to learn the next level on the entire skill challenge. Teachers need to be at least one instrument level higher than the PC.
I would suggest that if a PC is practicing in her spare time, she could take a maximum of 1 skill challenge per 2 weeks of casual practice. If she is actively spending lots of time learning the instrument, she could take a skill challenge every 3 days.