Injuries matter. Avoid things that are proven likely to hurt you, such as falling down this hill. Even a single broken bone can make you near useless for taking care of the littles, compare the price of two months of daycare vs solving the problem.
Powered mowers are ruled out. Walking on the hill without traction spikes is too. If you must do it in the short term, get kahtoola microspikes for traction and use the string trimmer or reel mower, its less likely to amputate you.
The real answer is to invest in a solution to keep you from mowing that steep hill. Replant shorter cover, starve it for water, cover in stabilized rock, or terrace it. If you terrace well, you now have a great vegetable garden area, and a selling point if you own the property. Keep the terraces under a set height (forgot what) and professionals and permits are not typically needed.
Maybe even consider deliberately killing the grass until you can solve it for good.