Ambiguity, ambiguity everywhere...
There are, as I see it, three questions here.
1. "Assuming each key is equally likely to be chosen, and that the bag is indeed shuffled after every trial, what is the probability of picking the correct key on the Nth try, assuming that you have failed N-1 times?"
Answer: 1/10, although this is by definition, since this is describing a Bernoulli trial.
https://en.wikipedia.org/wiki/Bernoulli_trial
2. "Prior to starting this experiment, what is the probability that it takes N trials to collect the key? What is the greatest value of this for a given N?"
Answer: p=(9/10)^(N-1)*(1/10), maximal at N=1. Again, this is more or less by definition, since we have set up a geometric distribution.
https://en.wikipedia.org/wiki/Geometric_distribution
3. "What is the expected number of tries in order to collect the key, given that we are allowed as many tries as possible?"
Answer: 10 (or more generally 1/p for probability p of success). This can be intuitively seen as bobbinwales described, or proven using the trick I hinted at.
TTT presumably meant to ask (2), but it's clear that (1) is also a possible interpretation.