The time to do something about that non-bloomer was a long time ago....when it was just a tad....3...4 years old.
Any peony that doesn't bloom after 3 or 4 years is thought will NEVER bloom unless you take steps to rectify by lifting and DIVIDING it.

So if you have one giant clump of a plant, do think MANY...clumps that can come from it as long as each has 3 - 5 eye sections and the parts are then planted at the CORRECT depth.....the topmost buds, be no more than 2"....
two inches....below the surface.
Deeper than that, the plant may resume its bad habit and NEVER bloom.

NEVER topdress a peony. We all do it; in early spring we like to reinvigorate our plots by topping it with some topsoil or compost we treasure. That's fine....but by putting soil over the base of the peony you are effectively putting it deeper....and in doing so may end up causing it to not bloom. Sidedress a peony only.

jeannie7....that some good advice. Thank you for that.

My longest is 7 yrs. I bought a tree peony in bloom, planted it in what appeared to be a good location. Every year it grew, nice, full leaves. No buds. Good dirt, all the plants, bulbs, trees around bloomed and did well. Finally last fall, I had a little talk with the tree peony. Said "Enough is enough. Either bloom next Spring or you are compost. I don't need a cute bush, I want flowers."

For what it's worth, that tree peony has about 8 buds on it now. Not ready yet, but soon. I really would have pulled him up this spring without flowers.

I also would divide the peony originally mentioned, make sure it is not too deep, in a sunny, well-drained location. Leave the bud eyes sticking out of the ground, cover with chopped leaves in fall. Tell the plant flat-out, he better produce flowers next spring. If he doesn't bloom next spring, off to the compost pile. Just because something is old, doesn't mean it EVER was a great plant. They had failures back then too. Bloom or toss it.

Then plant a good peony that will bloom, so you can enjoy it. We split my mother's peonies last fall, all are sending up leaves now. One is blooming the prettiest, almost coral color flowers. The others should be doing equally well soon. Shouldn't take forever either, to get flowers going. Good peonies produce good flowers, not headaches.

Thank you for the comments, and I'm still watching the one lone bud that seems to be growing, but very slowly and that could be in response to some very up and down temperatures, so it's not been consistently warm enough for faster growth. I must admit, I'd guess the plant in question had not even flowered for many years after a nearby very large tree put it into heavy shade most of the day, and of course it hadn't been watered or fed in years. Unfortunately as I moved it to a sunnier location, it must also contend with some very thick invasive tree roots where it's now growing , so it seem there's almost nowhere in my entire yard I can avoid those nasty and very overwhelming tree roots these days. I have been very careful with the planting depth etc. so that may not be a problem, but as for dividing the plant in the first place, it's a case of that poor plant being in such a poor state of health , that the youngest rhizomes were very tiny indeed, with the older parts fit only for disgarding, since they were full or holes etc and perhaps long dead, so it was also a case of having to regrow a good strong rhizome once again, with no good divisions possible at the time. Oh well, but I had the time and space wasn't an issue, though something else more immediately rewarding would have been better taking up the space. I'll keep you posted, if indeed the thing does reflower after perhaps over thirty or so years without a bloom! I think in many cases and in general , folks who plan to stay put in one house for many years, and who have young shrubs and trees, fail to take note of the decreasing amount of sun over the years as many of those sun loving plants end up shaded by ever increasing and spreading trees etc. I also have an ancient old peach tree or should I say the sucker from the mother plant, now in complete shade , perhaps thirty or fourty years after it was planted in what then was probably full sun!