We had Johnson Grass in one section of our lawn in Utah. The only way we finally got rid of it all was to mow, then let it grow so that the Johnson Grass jumped up much higher than the lawn grass. Then my husband and I would go out on the summer evenings, sit on the grass with a drink and pick at it until there were no more blades of tall Johnson Grass showing. We did this each time we mowed in the summer for at least 3 or 4 summers until finally no more tall blades appeared.
Rashel, in your asparagus bed, I think the only safe way to eradicate it is to dig a small section of the bed out at a time - and dig deep! - sift or otherwise separate it so you can pick out every single Johnson Grass root, then re-plant the asparagus. Be sure not to let the asparagus roots dry out before you get them re-planted. Maybe put them aside wrapped in some old wet towels or something?
You might lose one year's production on the asparagus this way, but you won't lose the whole bed. If you maybe do half the bed this year and half next year (and put a barrier between the weeded section so the grass doesn't invade again) you'll still have a little asparagus to harvest. I would think you'll have some nice big healthy asparagus roots after 5 years and they won't be set back all that much by being dug up and re-planted. Plus it will give you a great opportunity to amend the soil in your bed.
You certainly will lose production, and eventually lose the whole bed if you don't get rid of that accursed grass.
