"This assumes that the request is in God's will, which I think is a safe assumption for healing (?)."
This is a safe assumption based on the life and works of Jesus. Jesus is the exact representation of the Father and as such revealed the Father's will to heal in never refusing to heal someone who came to Him and never suggesting that someone needs to wait for their healing.
Bill Johnson expresses this idea like this: "Jesus is perfect theology".
This "healed all who came to Him" experience is not unique to Jesus. In Acts 5 we read of the Apostles performing many signs and wonders ... then in v16 we have " Crowds gathered also from the towns around Jerusalem, bringing their sick and those tormented by impure spirits, and all of them were healed." Also later in Acts we see Paul heal all on Malta.
It has been said that you can't pray a prayer of faith unless the will of God is known. So, knowing that it is God's will to heal in a specific healing prayer situation is important. All the healing evangelists that I have read about have this issue sorted - they believe it is God's will.
A prayer that says: "If it be Your will heal Bert ... " is not a prayer of faith.
There is another dimension to healing - as well as faith and God's will is the idea of authority. When Jesus sent the 12 and the 72 He gave them authority (not faith) before He sent them. At the Great Commission Jesus says "All authority is given to Me. Go, therefore ...".
So to answer your question, with a little modification: yes - if there is faith (which comes from knowing God's will to heal) and prayer is made in the authority that a believer has then God promises to deliver.
All that said, one of the first times I saw God heal when I prayed for someone I was rehearsing in my mind what I would say to explain why they hadn't got healed - and then the healing came. Where was the faith in that situation? I have heard that John Wimber had a similar experience - after praying for about 1000 people he was praying for another and rehearsing how to explain why the prayer didn't work when the healing came. (Not that I'm comparing myself to Wimber).