Moisten the flower bed with 1 inch of water. Pull and dig up the ivy the next day, ensuring that you remove and discard all the plant parts. Any ivy pieces left behind can take root and grow again. Loosen the soil using a mattock, if desired, but remove all the chopped roots. If the ivy returns, persistently dig or pull it out so it does not come back in full vigor.
Trim the ivy 2 or 3 inches above the ground with a weed trimmer or pair of pruning shears in fall. Paint the cut stems immediately with an herbicide that contains 41 percent glyphosate, or a brush killer that contains triclopyr, using a paint brush. If new growth occurs, wait until it is at least 6 inches long, then spray the foliage until it is moist with a ready-to-use (RTU) glyphosate product.
Spray a ready-to-use herbicide that contains 2,4-D and MCPP or glyphosate to moisten the foliage in spring when the vine is growing actively, or when it is in bloom, depending on the plant. Reapply the product in fall in a similar manner, and again the following spring, if necessary. You can also use a natural product that contains pelargonic acid or vinegar in the same manner if desired, but it likely will need more reapplications for the ivy to be eradicated from the flower bed.