How does one purify himself for prayer if three fingers of his right hand have been amputated, and there is a medical bandage on it from the beginning of the palm to the middle of the forearm, and his left hand is covered with bandages from the end of the fingers to near the elbow, so that only the knuckles of two fingers are visible on the right hand, and only the fingers on the left hand?

If you are unable to perform ablution (wudu') yourself, and you find someone to help you perform it, even for a fee, then you must wash your face, rinse your mouth, sniff water into your nose and blow it out, wash the rest of your visible limbs, and wipe over casts/bandages and your head. If you are unable to find someone to help you perform ablution, and you fear the prayer time will expire, and you are able to perform tayammum (definition) (dry ablution) yourself, then perform tayammum. Otherwise, seek someone to help you perform tayammum. If even that is not possible, then pray in your current state, and the obligation of purification (tahara) is waived for you, as Allah Almighty says: "So fear Allah as much as you are able."