Answer: Based on your description it sounds like there is a combination of problems going on. Tree roots combined with a slope can make growing anything a challenge, the soil is likely to be eroded, poor, and dry and the tree will outcompete just about any other plant for light, water and nutrients; it also blocks rain from reaching the ground. So the shrubs may be failing to thrive due to a variety of reasons, not just the ivy.
You may want to encourage the ivy to serve its purpose as a groundcover there because it sounds like such a difficult growing location. To help matters along, you might want to try to remove the weed barrier fabric as it can entangle tree roots as well as weed roots and does not prevent new weeds sprouting in rotted down mulch. Then hand pull/dig or spot treat the worst weeds; eventually the ivy will crowd them out naturally. Continue to mulch the area with organic mulch until the ivy fills in. The mulch will help keep down annual weeds and limit erosion as well as help feed the soil slowly over time as it breaks down. Rake or fluff and top up the mulch as needed to keep it about three inches thick; this prevents new weeds sprouting in the mulch. If weeds are really a problem, cut them off short, cover with a layer of damp newspaper and then top that with mulch. The paper excludes light and prevents growth. It will break down eventually beneath the mulch, as long as it is in direct contact with the soil, and the ivy can grow through it.
Hydrangeas need a soil that is organic and humusy and rich. Generally, they do not do well in competition with tree roots. The soil for hydrangeas must also be evenly moist yet well drained and most slopes are just too dry for them to do well. So, I would suggest you look for a different spot for your hydrangeas. Ideally: morning sun or bright dappled light all day plus good protection from winter winds.
As far as the impatiens, you probably will not have very good luck planting them directly in the ground, but you could put them in large containers by the walk. This would give you a high impact color statement all summer and if you use attractive containers, a nice visual effect in winter as well.
I am sorry I can't be more encouraging about what you originally intended, but I hope this gives you some ideas.
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