We keep mowing and burning. Burning is not an option for you.
After five years your asparagus bed should have multiplied nicely. Do you have any place to relocate it? If you do, I would suggest digging it up and separating it carefully and get every bit of Johnson grass root out. Sometimes flushing in water will help but the asparagus roots will be dense. We had to move our asparagus after two years and it multiplied so much we cut each crown into at least four sections and replanted. Never missed a beat the next year.