Hello,
It looks like the nerves of the hand had been damaged due to the bite(if severe) or there is a continuous irritation of nerves. But to rule out all other medical conditions causing numbness/tingling like diabetes, peripheral neuropathy,hypothyroidism,vit B12 deficiency ,autoimmune or neurological diseases which may cause the same, I would suggest you to consult a neurologist and get MR neurography and nerve conduction velocity tests done. Treatment will depend on the confirmation of diagnosis.
I sincerely hope that helps. Take care and keep me posted Kind Regards.
My goodness, you have my sympathies. This is one of the reasons I devoted a specific section to killing rats in the emergency preparedness section. They are not a "minor annoyance".