A few suggestions, having gone through this process myself recently with a second hand 2016 30kWh Leaf.
First of all, you need to create a You+Nissan account through the website first before you can use the App, so make sure you do that first.
Secondly, you don't add the VIN in the app, you add it in the You+Nissan account web portal. Until the car is added to your account through the web portal you won't be able to log into the phone app. When you try to add the VIN it will ask you to follow some steps to turn the car on and off again - this process is slow and quite flakey - it took me several attempts to "verify" the car.
What model and year of Leaf is it ? There are two different apps to use depending on year. For cars up until about March (??) 2018 you should be installing "Nissan Connect EV". For later cars you should be using "Nissan Connect Services". (There is a 3rd app which is for the US market, don't try to use that one)
If you're still stuck, phone Nissan technical support via the number on the website. They will open a ticket to have the car added manually to your account for you - I had to get this done as the car was still on the previous owners account. To prove I was the new owner I had to scan and email copies of my drivers license and my new V5C.
Nissan support are SLOW - it will be about 4 working days between each contact with them before they respond to your previous email or phone call. So count on a week or two to get this sorted. You're best to make initial contact via phone so there isn't a 4 day delay before they ask you for initial details, then the follow ups will be by email. Be patient and they will eventually reply.
As for the app not being very good - yeah, it has its issues and quirks, but it works pretty well for me for starting/stopping climate control remotely, checking charge level and charge time and remotely overriding the charge timer. I think it's worth it and I'm glad I have it set up.
Another tip is once you have it set up look at the car in your You+Nissan portal - it will probably say that you are "subscribed" to 3 years of free Nissan Connect services. I had the additional option to "subscribe" to 8 years of Nissan Connect services for free, so naturally I did that, so that lasts until 2028.
Not sure if that is a promotional deal or standard, but if you see the option to extend the free period for free, do it in case they take away that free subscription extension some time in the next 3 years...
