While it may be too small for your filters there is another issue filter wise. For any mechanical filter to work there must be some way for debris to be moved into the filter. Most dead algae settles on the bottom so a mechanical filter wouldn't have a chance.
Depending on the algae species and your filter it could have be large enough for the filter when it first died. But once dead on the bottom it decomposes into smaller bits.
There are different oxidizers you can use, chlorine, hydrogen peroxide, potassium permanganate, etc. These break muck into various things including some gases which with strong agitation can out gas a fair amount of the muck as a gas. The downside is these chemicals can certainly kill fish so a person really has to know what they're doing. No, seriously know what they're doing. Not the standard internet oh I'm a really smart blowhard type. Other downsides include being very expensive and very time consuming because oxidizing the amount of matter we're probably talking about is a lot and would require many applications. People who use these do so on a regular basis when the pond is what most people would call "clean". So they only have to oxidize a small amount of matter. And they know what they're doing. They've done more research than reading a post from some dude that says "dump this in, it worked for me".
There are bottles of stuff that you can buy which promise to make muck disappear, or "clean pond" or "clear water". Generally they say something about having bacteria and skip over that their bacteria and the bacteria already in your pond are the same thing. They don't work at all, but enough people "see" them working that these products stay alive. You can try some if you like, not too expensive. They do add more dead organic waste to your pond but, but only a small percentage more compared to what's already there, so not a huge downside.
I would guess there are probably lots of other products being sold that promise a "clean pond" or "clear water". They come and go. Same manufacturers, they just have to keep changing the names and packages as people catch on to the scam. Hard to keep up with them all, which of course is the idea.
For ponds built without automatic cleaning systems like bottom drain-TPR-sieve, the easiest and most effective removal method is vacuuming. You can hire this out. But most companies doing this type of thing drain the pond, power wash and shop vac out the muck. They do this because they're geared to a particular type of pond, ponds with loose rock. Vacuuming isn't very effective when there's loose rock.