The fastest way to stop hunting for keys is to drill one row at a time until the whole row is automatic, then move on. Pick a set above, type the romaji for the kana on the board, and the grid tracks where you are.
The rows here are the same ones on the romaji reference page, generated from the same engine — so the spelling you drill is the spelling the game accepts, including the l/x forms for small kana and the hyphen for the long-vowel mark ー.
This trainer is romaji-only by design. An IME would convert the kana away before it could be matched, and typing them directly is the entire point.
Next steps: the full romaji table · take the typing test · play the Yamanote Line