You need to tell your manager that it's simply impossible to both clean all that up and also make progress on the project. You need to stop the erosion in order to fix it; fixing will require fundamental changes that will absolutely block progress. If aren't able to freeze the project, your cleanup progress will be constantly thwarted by new ugly hacks.
There's just no way around it. If your managers won't freeze the project until you make meaningful progress, you're going to be stuck working with this ugly code for a long time. Try to convince them that in the long-term a cleanup will pay dividends. Without their support though, you won't get anywhere.
If you're able to freeze the code for a bit, it sounds like it might just be quicker to rewrite some parts like the CSS. Don't take on the whole project at once - find pieces that can be rewritten individually, even if it's a large part like all the styling.
There's just no way around it. If your managers won't freeze the project until you make meaningful progress, you're going to be stuck working with this ugly code for a long time. Try to convince them that in the long-term a cleanup will pay dividends. Without their support though, you won't get anywhere.
If you're able to freeze the code for a bit, it sounds like it might just be quicker to rewrite some parts like the CSS. Don't take on the whole project at once - find pieces that can be rewritten individually, even if it's a large part like all the styling.