I use a weed whacker to cut up the ivy, but not take it to ground level. You want to just barely cut the leaves. Then apply either Round-up or a specific ivy killer. The weed whacker cuts the leaf and stems to allow the chemical to enter the leaf and stem and kill the plant. Ivy is a tough plant with a coating on the leaves and stems that is difficult for chemicals to get through without this "pre-treatment".
The same treatment might work on the bamboo if you repeat it several times during the growing season.