You cna approximate this via the Central Limit Theorem or chebyshev's.

This is a multinomial, you can calculate the mean and variance easily.

The question seems to be, calculate n and I'd use the CLT.

where is the number of letters she receives on day i.

and each has that distribution you stated

Next calculate the mean and variance and you can divide by n if you wish........

and you may want a correction factor adjustment of 98.5 here

You need to look up the normal percentile and then you solve for n, if I understood this correctly.