"favorably impressed" is correct and usable in written English. You can use it to describe how someone feels about something they have seen, experienced, or heard. For example, "I was favorably impressed by the performance of the orchestra."
Paul Koehler of RPGamer, writing up his impression of the game at TGS 2007, was favorably impressed, calling map navigation "fluid" and saying that the graphics were "one of the better offerings at the show on the DS".
The girls were not favorably impressed.
But the regulators were not favorably impressed.
Everybody was so favorably impressed that CBS Records signed them.
Akumal was favorably impressed with what he saw.
A.: I was quite favorably impressed by him.
But Goldman is favorably impressed with Mr. Siewert and the discussions are continuing.
He took four shots and favorably impressed Tortorella by "playing within himself".
But most Americans said they were favorably impressed, at times even emotionally moved, by Mr. Bush.
Mr. Cohen said he was "very favorably impressed" with Mr. O'Neill.
Mr. Ishom, who speaks and writes English well, was the most favorably impressed with the program.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union