It already gives "He will be well quickly" and also claims that "well" and "healthy" are both possible translations of "gesund" in that context and so "He will be healthy quickly" should definitely be correct.
In German the present is used instead of the future in many cases. Here the complete form should be "Er wird schnell gesund werden." I translated it as "He will soon recover". It resulted in a mistake, but I reported it. :)
The correct solution is suggested as :
• He gets soon healthy.
• He becomes healthy quickly.
English is my mother tongue and I don't think I have ever heard these words spoken. There has to be a better translation.