I would just put the fertilizer on/around plants that need it. I've never heard of anyone fertilizing compost before. I would chop up the roots and throw them back into the compost, or may just spread them around the surface of a cultivated area to be part of the mulch. After baking in the sun, they will be killed.
Composting is not a specific recipe that needs to be followed, just a method of retaining home and yard waste to use as a positive. If it rots, it can be composted. If you're worried that it's not dead enough to compost yet, you could steep it in a bucket of water until it turns slimy, or enclose it in a plastic bag until the same thing happens. Then make a little hole in the compost and put it in. Or things like that can be baked in the sun until surely dead, but they don't make as active of a contribution to the compost if processed that way. Chopping things into smaller pieces is also an option though I wouldn't trust pieces of ivy, in particular, unless I was sure they were dead first because one could just end up propagating that instead of composting it. (Assuming Hedera helix.)
I don't think you're going to get heat steaming from a pile that small. Heat is a very over-hyped aspect of composting that home composters often get hung-up on. For the past 20 yrs, I've been cold composting and there is no difference, except time, from the 1 time I built a pile big enough to get hot (several cubic yards of material).
It's also not necessary to compost organic matter first, if you're not having fun with the compost pile. OM can be slightly buried between plants, or hidden behind them on the soil surface. During the heat of summer, most matter will lose recognizability after just a few days and look like mulch. One can dig a hole and fill it over the course of a week, cap it with some soil or dry leaves, move to another spot for next week's hole.