It means "I can write very much better than he can." And the subtext, probably, is "So give that large contract and advance payment to me [and not to Jack]. "
It mixes a couple of fairly child-like idioms to achieve this. "I can do it with my eyes closed" is a fun, or boastful, or childlike way of saying that I can do something very easily. And "you can play somebody off the park" is from the world of sports, where it is used to mean that you are a very much better player than the other person.
The writer here has combined these two usages. " I can write him /// adapted sports metaphor - "off the page" //// childlike boast "with my eyes closed."
I hope this helps.