Health insurance for students in Germany is roughly between 110-120€ per month, so your bill would be for a period of around 9 months, is this so?
Once you stopped receiving asylum benefits, you are responsible for paying for your health insurance. If you had received Bafög, this would have included money to help cover your health insurance. Otherwise, unfortunately, you have to pay. I advise you to check the amount they are charging you: is it the correct rate (i.e. the reduced monthly for students) and for the right number of months? Possibly you can have the amount reduced a bit. You can ask to pay in installments, this is called Rantenzahlung. To do this you make a repayment agreement with them called a Ratenzahlungsverinbarung.
As you write that the issue with Bafög was due to a deadline being missed by the Ausländerbehörde, I recommend you also looking into doing what’s called an ‘Überprüfungsantrag’. This allows you to have a past decision made by an authority checked again. If it can be shown that the Bafög application was rejected for reasons beyond your control (i.e. due to the amount of time needed by another authority), then you can still be granted Bafög for the period concerned.
I hope that helps clarify the situation a bit.