Removing a penny raises mean coin value from 17 to 18. How many nickles? So this is a question that my sister (grade 8) got wrong on a test.

There is a collection of quarters, dimes, nickels, and pennies in a jar. The mean of these coins is 17 but when you remove a penny, the mean becomes 18.
How many nickels are in the jar?

I tried to get the multiple of $17$ which subtracted by $1$ and divided by $16$ is $18$.
For that I got $17 \cdot 17 = 289 - 1 = 288 / 16 = 18$ which is correct.
The only problem that I'm encountering is which coins are in the collection and of course how many nickels are there.

You got most of the way there already. You know there are 16 coins that are worth a total of 288 cents. Three of these must be pennies (since 288 is 3 mod 5 and all non-pennies have value 0 mod 5), so that leaves 13 worth 285. 
This can be achieved with 11 quarters and 2 nickels.
In a jar with $Q$ quarters and $13-Q$ other coins, for $Q\leq10$, the total value of the jar is at most $25Q+10(13-Q)=130+15Q\leq280<285$ cents, so you need at least 11 quarters. However, with 11 quarters, you have ten cents left and 2 coins, so they must both be nickels. 
