Health Insurance Backcharge

It is compulsory to have health insurance while living in Germany. Therefore your membership of TK did not end, or become free of charge, just because you lost your job. You will have to pay the outstanding fees - there is no way around that (short of declaring outright bancruptcy).
The only question here is how much you need to pay. If you have no employment income and don't receive unemployment benefits or social security payments (of which you didn't mention anything), then you are classified as self-employed (or rather, self-financed) an then the rate is between EUR350 and EUR750 per month, depending on your actual income. Since TK had no information about your actual income, they "estimated" you into the highest bracket. You should appeal against that and submit proof of your actual financial situation - if you are lucky, your debt can thus be reduced to EUR2800. They are also usually open to deferred payment in installments (as long as from now on you pay your regular monthly due on time).

Thank you lot for your input. I will update this thread if I get an response from TK. I have mentioned the reason why I was in such a situation. I hope they will understand.

If you had applied for unemployment or social security benefits, the health insurance would be free of charge.
You are entitled for unemployment benefits if you were previously employed (with compulsory unemployment insurance contributions) fo r at least 12 months and are looking for a new job.
Social security benefits are more complicated and, since you are non-EU-citizen, most probably don't apply for you (but this depends on your visa and how/why you came here).