Shouldn't be kindergarden?
According to English Wiktionary, 'kindergarden' is a misspelling of 'kindergarten'. The word is a borrowing from German and came to English as a whole, without changing.
Agreed. "Kindergarten" is the standard English word.
I do not understand, ta przedszkoła is a female gender, why - w tym przedszkolu?
It's przedszkole and it's neuter.