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If your Ledger screen is terribly dark but still displays something very faintly, have you tried to amplify that by looking at the screen with a mobile phone camera? I've seen some examples of this intermediate partial remedy of the dying screen issue on Youtube. Try this in a dark environment, of course turn off the mobile phone camera's flashlight.
Maybe worth a try to move your coins from the Ledger wallet if the view by mobile phone camera makes the faint display visible for you. As others said, nothing wrong to use your KeepKey as recovery target device for your Ledger wallet, unless you can't recover all properly.
When you were able to transfer your coins to a new Trezor wallet, do not throw away your recovery details of your Ledger wallet. Do not use the Ledger wallet any further, but keep the recovery details with lower storage security, just in case you need it for whatever purpose may surprise you in the future. (General advise: never completely destroy a wallet you ever used. You never know when you might need it ever again and it doesn't hurt to keep it at low storage security.)
I've actually tried both, using my phone camera with and without the flashlight and opening it in the dark, all to no avail. That's why I finally decided to make use of my unopened KeepKey. This is all temporary anyway, while I'm waiting for my Trezor. But I don't mind leaving little amounts to both my KeepKey and Ledger if I finally get the screen replacement. God forbid, but should a $5 wrench attack happens, these 2 might save my main wallet.
