(Keeping in mind I haven’t read the material), ちゃう can be used for other meanings as well. Here I think it’s more regret, like, “I’ve been busy with this weird case, you see? Just can’t figure it out.” He really doesn’t want to be stuck on it, but it’s a strange case that he just can’t figure out. Or, more specifically to the actual line here, “I’ve been caught up in this weird case, and it’s kept me busy.” It’s a bit hard for me to translate the ちゃう into English, though… (I tried by writing かかわっちゃって as “caught up”, wording which in English can imply being stuck on something one doesn’t want to be part of).
Here’s a good video on it. You may have already seen it, in which case, it can be a refresher:
Since this isn’t a request (one reason to end in て), I get the feeling like he’s trailing off a bit. Like, he has more to say, but he just kind of ends it there. “I’m caught up on this weird case, and it’s kept me busy…you see?”