What's happening inside the Old Lady?

"The house that Jack built" is said to be a cumulative tale

In a cumulative tale, sometimes also called a chain tale, action or dialogue repeats and builds up in some way as the tale progresses. With only the sparest of plots, these tales often depend upon repetition and rhythm for their effect, and can require a skilled storyteller to negotiate their tongue-twisting repetitions in performance.

Wikipedia also lists "There Was an Old Lady Who Swallowed a Fly" and "The Twelve Days of Christmas" as further examples.

In folk music circles songs like this (for example The Twelve Days of Christmas) are known as "cumulative songs" or "accumulating chorus songs". Accumulating chorus songs are slightly different from "There was an Old Lady" in that they tend to have verses each of which stands on its own but adds its item to the chorus which thus grows longer as the song progresses.

Not all accumulating chorus songs are traditional, or even that old. The late Keith Marsden wrote one called Doin' The Manch in, I think the 1970s. It involves a lad being taken by his father, on his eighteenth birthday, up the roughly mile to a mile and a half of Manchester Road in Bradford to drink a pint in every one of the twenty seven pubs that used to line it. Most verses add a pub or two to the chorus and one adds the names of a two breweries into the mix.

It's well worth a listen, find it on You tube

There is the law of holes with the first being "when you find yourself in a hole, stop digging." Since you mention it in your question, that's the adage about the action. It's common enough to say "Remember the law of holes," or for your example, "It seemed simple enough at first, but before long we forgot about the law of holes."

A vicious cycle / circle:

A sequence of reciprocal cause and effect in which two or more elements intensify and aggravate each other, leading inexorably to a worsening of the situation.

https://www.oxfordreference.com/view/10.1093/oi/authority.20110803115639406

I appreciate this isn't the specific term you're looking for (which I think probably doesn't exist in English, unfortunately), but I would naturally just describe such a situation by saying everything we did made things worse, or every fix just introduced new issues (which sounds quite techn-jargony) or similar.

The common phrase is just

bad to worse

We kept going from bad to worse!

It doesn't explicitly have the "cumulative" you're after but that's about the closest.

Another one that you hear is "we just kept digging ourselves deeper and deeper in a hole..."

I would use "compounding the problem" but as OP has used "compounding" a couple of times that probably isn't what they are looking for. But I still wanted to suggest it for other people.

This doesn't exactly capture the compounding nature of the situation, but I came across the term "solution-caused problem" used in an article by a consulting firm to describe problems caused by the solutions to different problems.

Alone it might not fully express the idea, so I would add something about the recurrent nature of the problems, like a loop:

We are stuck in a nightmare loop of solution-caused problems.

A term that may fit is a cascading failure:

A cascading failure is a process in a system of interconnected parts in which the failure of one or few parts can trigger the failure of other parts and so on. (Wikipedia)

An example would be one component overheating, which overloads a cooling system, in turn causing other components to overheat which would then trigger failures in items depending on those components, and so on.

I've heard plumbers complain of "snake joint jobs." You go to change a washer, but the screw strips, so you have to replace the core, but there's too much calcium, so you need a new faucet, and when you loosen the nuts, the pipe breaks off inside the wall, so you get the water shut off, but not before a good amount of water damage happens. Then you tear into the sheetrock....