Your understanding is basically correct.
As a pattern, it is possible to think that 何/どの/どこ etc + ても/でも as no matter /...ever, which means roughly the number does not matter (in your case the number of nights). Literally your sentence means no matter how many nights you stay, please (stay).
- 何度やってもうまくいかない No matter how many times I try, it does not work
- 何回聞いても忘れる No matter how many times I'm told, I forget it (easily).
- どんな手段を用いてもかまわない I don't care no matter what means you use.
- どれだけ寝てもまだ眠い However much I sleep, I still fell sleepy.
- どんな猫でもかわいい Whichever cat it is, it is cute.
As seen in the last example, it is not always natural to use no matter/...ever, it would be more natural to use as long as or simply any.