Thanks for posting your query.
I have seen the photographs posted in your profile.
From the history that you have provided it looks like paronychia of the nail which progressed to infection of the fingertip. Paronychia is a soft tissue infection around a fingernail that begins as cellulitis but that may progress to a definite abscess. It is of two types: acute caused by bacterial infection and chronic caused by fungal infection.
From your sister’s symptoms, the hangnail might have started the chain of events by causing first paronychia which then progressed to infection of the fingertip causing a felon. A felon is a fingertip abscess deep in the palm side of the finger. It usually is caused by bacterial infection, but a herpes virus called herpetic whitlow and, more rarely, fungi also can cause felons. In herpetic whitlow, small bubble-like cysts on the skin, called vesicles, and repeated episodes are there which distinguish it from bacterial or fungal infection. It is less likely to be the cause in your sister’s case. Hence it can either be bacterial or fungal infection.
Your sister has taken many antibiotics and antifungals but they have provided little help. In that case, the best treatment is incision and drainage followed by a round of general antibiotics which in turn is followed by specific antibiotics or antifungals after the reports of pus/ blood sample of the effected site specify whether it is bacterial or fungal infection.
Incision and drainage for a felon is an OPD procedure in which the doctor makes the whole finger numb by injecting an anesthetic once into each side of the finger. Then he or she will make one or more small cuts in the tip of the finger to allow the pus to drain out. A narrow gauze strip may be left in the wound to hold it open for continued drainage. It is followed by broad spectrum antibiotics and sending the pus sample to the lab for bacterial and fungal culture examination. Once the culture reports come, her doctor will prescribe specific antibiotics or antifungals.
If it’s not treated, a finger pulp infection can cause a number of complications like spreading of the infection to the bone of the finger causing osteomyelitis, skin necrosis, septic arthritis and tenosynovitis (when the infection spreads to the sheath that surrounds the tendons in your finger). Hence my sincere suggestion would be to get it evaluated from a specialist and get the above mentioned treatment.
Hope that this information helps and hope that your sister will get better soon.
Wishing you good health.
Attention: I uploaded the relevant photos to my profile. Please try to view them.