Revealing original message length having random padding

If I have a message that has a fixed unknown length $L$, and we add to it's TLS encryption a random sized padding $0\leq n \leq N$ so the sent message if $L+n$. I'm also able to make the target re-encrypt and send the message over and over again.

How many times do I need to make the target send the message over and over again, until I reveal whether the original length is $L$ or $L+1$?

I think that I need to make it send it at least $N+1$ times, so i'll "cover" all of the options of the padding length, but I don't know how to continue from here.

Assuming that the message is known to be of length $L$ or $L+1$ and that the padding length is uniformly distributed, then unless the cipher is of length $L$ or $L+N$ both message lengths are equally likely.

The question then is how long do we have to wait to see a cipher of distinguishing length. As noted in the comments, this is a Bernoulli process with parameter $1/(N+1)$. As such the waiting time for a success satisfies a geometric distribution with parameter $1/(N+1)$.

Your intuition is correct that the mean waiting time is $N+1$, but you should beware that the geometric distribution has a heavy tail. For example, the chance that you might need to examine $2N$ or more ciphers is about $e^{-2}\approx 0.135$ which is quite large. If you wanted to be 95% sure of seeing a cipher of distinguishing length, you might need to look at $3N$ or more ciphers.