Hoeing is really only for controlling annual weeds.
Well-established (or invading-from-adjacent-lawn) perennial weeds, particularly ones that spread from runners or bits of root, are best controlled by hand-pulling the plants with as much roots as possible still attached. (Easy for some things, harder for others, but ya do what ya can).
So first remove all the root material you can possibly get out. If there are plants already planted in the beds, so you can't just fork the whole bed over and pick thru every bit of it, it can be useful to have a long screwdriver or other similar tool so that you can stick it underneath much of the plant's roots and pull up with the tool held semi-horizontal as you also pull the plant, so fewer roots break off in the soil.
AFTER you've done as much as you can that way, THEN you can use a (sharp!) hoe to lop off new topgrowth that appears, ideally at least once a week. This will help starve the remaining roots to eventual death.
But just trying to hoe a running perennial to death (like any of the weeds that you could possibly be referring to as ivy ) BEFORE removing all possible plants/roots is at best going to put you on a perpetual treadmill (good exercise tho!) and possibly not succeed even then.
Mulching over a strong stand of invasive-rooted weeds actually tends to make them WORSE, they pop up thru it and it keeps down their competitors
The best long-term solution is to wait til the garden is done for the year, this fall, and then fork it ALL over and hand-pick roots out of ALL the soil. At least for as large a portion of the garden as you can manage. That makes a HUGE difference in future years' maintenance!
My invader is Bermuda grass. I dig up the area around the edge of my garden in the offseason and remove as many roots as I can. Then around the inside of my fence I put cardboard down and cover that with mulch, either seasoned wood chips or grass trimmings. That gives me a reasonable place to walk without walking in my garden just to get from one side to the other. The cardboard keeps the Bermuda grass from poking through and the mulch holds the cardboard in place. It can be a little slick when wet, but nothing is perfect.
The Bermuda sends runners a few feet under that cardboard to try to get to my garden. I fight those as they come through to try to keep them from getting a foothold in my garden proper. I won't say I always win that fight, but I try.
Then, in the offseason, I turn that area under the cardboard and pick out the roots. That cardboard has usually decayed so much that I just turn it and the mulch on it over into the soil. That is great to go in the regular garden area, but be careful not to put any roots or grass with it. With Bermuda grass, all it takes to start a new plant is a half inch of root or stem, especially if it is at a node.
I probably did not mention, Bermuda grass is great in the lawn but I really hate it in the garden. These invaders that spread by sending out runners are a real pain. Edging does not do much good unless you really get it deep. I've seen Bermuda grass runners more than 8" deep. Edging does define a border though which can help.
Good luck with it. I don't have any easy solutions.