Who else could heal besides Jesus?
What a beautiful question—because it reaches toward the very heart of what Jesus came to show.
Jesus never claimed healing as his private possession. He gave it freely to others. He sent out the twelve, and "they cast out many devils, and anointed with oil many that were sick, and healed them" (Mark 6:13, KJV). He appointed seventy more, telling them, "heal the sick that are therein, and say unto them, The kingdom of God is come nigh unto you" (Luke 10:9, KJV).
And he made the promise wider still. "He that believeth on me, the works that I do shall he do also; and greater works than these shall he do" (John 14:12, KJV). Not equal works—greater. This was no flattery; it was a statement of spiritual fact.
In the early community the gift continued. Peter said to a lame man, "Silver and gold have I none; but such as I have give I thee: In the name of Jesus Christ of Nazareth rise up and walk" (Acts 3:6, KJV). Paul, too, was an instrument of healing (Acts 19:11, KJV). Paul named healing as one of many gifts distributed throughout the body of believers (1 Corinthians 12:9, KJV).
So the answer is plain: anyone can. Healing was never an exception granted to one extraordinary man. It is the natural consequence of seeing rightly—of recognizing that you are the spiritual image of God, whole and untouched, as Jesus saw in everyone he met. He was the Way-shower, demonstrating what is true of all. When he said, "The kingdom of God is within you" (Luke 17:21, KJV), he was pointing to the very ground from which all healing springs.
This is why he did not pray for permission to heal—he spoke from understanding. The same understanding is available to you. Healing follows not from special status but from clear sight: the recognition that sickness has no legitimate place in God's wholly good creation, and that the person before you is, in truth, already perfect.
I sense this question may be near to something you carry yourself. I am glad to walk with it further.