とにかくあの怪獣犬から少しでも遠くに離れたい一心だ。かけにかけ続けて、やっと七階の太ったレディの肖像画までたどり着いた。
Anyway, they wanted to get as far away as they could from the monster dog. ???? and finally made it to the portrait of the fat lady on the seventh floor.
かける is a word that seems to have a million meanings and idiomatic uses. I have no idea what かけにかけ続けて means here. I would guess the most literal meaning is "continue to hang on a hook", perhaps meaning they continued to be in jeopardy until they reached the 7th floor?? I also see that かけ can mean 'gallop', which may be relevant. All pure speculation. I cannot make a confident guess.
Edit The translation in A.Ellett's comment seems convincing. I guess then that this is 連用形+に+連用形 and is used for emphasis: "they continued to run and run". Could I equally well use 走りに走り続けて?