Hi,
I had very little knowledge or experience with cars, so the steps I mentioned earlier were as far as I could manage on my own. Fortunately, I found a mechanic who was able to assist. I’m not entirely sure how he managed to slide the core out, but he did, and he replaced it with the part I had sourced. That resolved the issue completely.
I believe I paid around £100–£200 for the repair, which was significantly cheaper than the quotes I had received from Audi and other garages.
From what I saw, a flush might have worked as well, as the old core was clogged with a lot of debris. If you’re facing high repair quotes, it might be worth trying that first.
Sorry I couldn’t be more helpful, good luck with your repair.