The solution is to have at least two piles. Since you are constantly adding, there will always be some percentage in the pile that is not going to be decomposed. If you stop adding to a pile once it is of sufficient size, then in a few months it will be properly aged and ready to go. It will also reduce the problem of having a pile that seems too large to turn. Your compost will also be much slower to decompose in the winter, cool temperatures slow down the activity of the microorganisms that are doing the bulk of the work, and if you aren't turning the pile the center will be well-started but the outside will not make progress at all.

For this year, start a new pile with anything that is readily identifiable, and do it as soon as possible. Separate what you can that is usable, ideally with a sifter, but with a fork or rake otherwise, so that you will have something to work with this spring and will reduce the size of the pile that you are starting with this summer. An effective way to do this would be to spread the piles on the garden as they are right now, then rake up the larger pieces, back off the beds and create a new pile with them. What gets left behind will finish decomposing in the garden, and should be small enough to incorporate into the soil without much difficulty, even if you let that be done by the worms and bugs that are waiting in the garden already. Don't bother with activators, just pull some weeds, keep some soil on the roots, and add them to the new pile, and that will introduce any microorganisms that already exist in your soil, as well as what the old compost materials bring to the mix.

Once your pile is smaller and more manageable, you will find it easier to turn, which will result in faster creation of usable compost. You should get to a point where you have useable compost every 4 months or so, with a large batch available in spring from a slow process over the winter, a smaller but much faster batch in late summer for top-dressing your growing crops, and a final batch in fall for spreading on your lawn and perennial beds and any fall-planted veggies.

The other thing that you can do right now is devise a way to shred the material that is still too large. This increases the surface area that is available to the decomposers that will be doing the work for you, and will speed up the process dramatically.

What he said.

For a cheap compost accelerator, use any high-nitrogen fertilizer, dissolved in water. A couple of cups of ammonium nitrate dissolved in a bucket or two of warm water, and dumped on the pile will really speed things up.

Keeping things moist helps.

Accept that the material will not be compost in a month.

My piles have core heat of 130 to 160F at least at some point, but it still takes about four months or more to get a decent product I might call '75% composted'. And that was with shredded leaves as my main 'brown'. This month I used a lot of sizable wood chips in my two piles for the first time in ten years, so the expected time frame is even longer.

I usually end up sifting contents of a pile thru a 1/4" mesh sifter to get some material early. The larger material goes back into the pile.

You could move some material into smaller and more manageable piles, or spread the material out into a shallow bed. And if necessary this year, buy compost from a local company that concentrates on compost.

You could get bins and then turn your compost, balance browns and greens to start hot composting. That will speed things up over a pile on the ground. Microwave or boil the food things like apple cores you put in or you can even chop them up, use a blender maybe. Get some starbucks used coffee grounds, add bagged browns. You don't have to do all of that, but some of them will speed it up. Cold composting takes 6 months to a year or even more if the weather is cold.

With the luxury of time and space, I produce a lot of "cold," infrequently-turned, year-old compost. In general, I prefer its' texture (all my compost gets sifted before application EXCEPT these) which are primarily for end of season topdressing prior to cover crops or as a mulch over beds that will be uncovered (raked off and returned to "hot" piles) and planted in early spring - or as soon as this snow is gone, whichever comes first. With limited space and time, the extra effort of hot composting is, IMO, worth it because of the increased production.

I added a couple of handfuls of red worms to my large (10 x 16), mainly static pile last spring and they are reproducing like mad. They have been active all winter - granted it was a mild one but still had many nights below freezing. When a worm is not reproducing it is makin' compost. My hot piles have all turned into cold piles with worms. My .02.

I'll say it:

Start urinating on your pile and turn it as often as you can :-)

Here's another option:
If you have an old plastic garbage can with a locking lid, you might try making a do-it-yourself rapid composter.

There's some great information about hot composting in the FAQ. If you take out the big pieces and heat up your compost, it should break down pretty quickly.

DIY rapid composter
Google "trash can composter" for proper instructions, but basically:
Use a big nail to make a bunch of holes on the bottom and on the bottom foot or so of the sides of the can. Shovel in the compost from your pile, removing any large chunks, (You can use those to start your pile for next year.) Lock the lid on the can.

Every few days, push the can over on its side, roll it around a bit. This will stir it up and keep it active.

If it doesn't get hot within the first 24 hours, some of the fertilizer water another poster suggested should get it going. (You'll know if you have hot compost--you'll feel the heat radiating off when you put your hand over it.)

I also agree with the previous poster that you really should be fine if you spread it out over the beds and pull out any obvious chunks of stuff that didn't break down. It's called sheet composting, and people do it all the time.
Good luck.

Here is a link that might be useful: FAQ Intro to composting

If the C:N ratio in that pile is not close to the optimal 30:1, evidenced by heat generated by bacterial activity, then the material will not get digested. If the moisture level is not close to about right the bacteria tht digest the material cannot function and the material will not get diested. If there is not enough air in the material for the bacteria that will be digesting it that material will not be ready.
Any heat?
Your material may need a Nitrogen source.
Your material may need mixing to provide air?
Your material may need some water to provide adequate amounts of, but not too much, moisture.
Your material may need all three.