Dear @Sidra,
First of all, I am so sorry to hear about your loss. May your happy memories give you peace and confort during this challenging time.
I can understand that it is even more difficult to deal with these bureaucratic issues now. To be honest, I never had to deal with a similar question. But I assume that the responsibility for the payment is dependent on the funding agency which was supporting your mother/is supporting you as her family. If you are receiving asylum seekers allowance, the job center is most probably right saying that the "Sozialamt"/Rathaus is responsible. If you hold an actual residence permit and get supported by the job center already, they might be the right address to apply for the reimbursement.
Can you add some information here dear @mbeon-Éanna?
All the best,
Meike