I'm trying to get the privkey with this information, but I think I'm missing something

Thanks for all the learning

Greetings.
Sure!

We know:
lambda * x = a [mod N]
x = a + 1 [mod N]
So:
lambda * (a + 1) = a [mod N]
lambda * a + lambda = a [mod N]
(lambda - 1) * a + lambda = 0 [mod N]
a = -lambda / (lambda - 1) [mod N]