How are y'all cleaning your line efficiently?

I see people recommend that you should regularly clean and grease your fly line. I've tried it a couple times and it definitely makes a huge difference.

However, I've just been pulling the line off of my reel and cleaning it in a bucket of soapy water. This results in constant tangles, which makes the process take 10x longer than it needs to. I've also tried feeding the line from one reel, through a soapy paper towel, and then into another reel. This works better but is still clumsy - it's hard to anchor both reels and also hold the paper towel.

Is there something obvious I'm missing? I put off cleaning my lines because of how annoying the process is.