The only practical way to do this is with fluorescent or LED strip. Non dimming, fluorescent will be the best value. A high color rendering warm lamp like an 830 series T8 would work well. Dimming fluorescent substantially increases the cost. Name the reason's you'd prefer to avoid fluorescent.
LED rope is designed to mostly be viewed. It barely makes a glow in a cove hidden from view. Higher power LED light strips like Philips Color Kinetics or Maxim can put out a lot more light but cost many times as much.

Here is a link that might be useful: Philips link

I've considered retrofitting a burned out lumiline lamp with a string of 10 12V 6W bulbs in series, perhaps tubular "festoon" bulbs commonly used in automotive dome lights. Another option would be strips of LEDs, but in my application the lamps are exposed so it would have to fit within an existing lamp tube, or a frosted plastic tube of the same dimensions. It's difficult to keep LEDs cool in an application like this and it would require an external driver.

I need a few of the 2 prong Lumiline light sockets and can't find them anywhere locally. Does anyone have a few lying around not being used?
Thanks

I noticed a few people here are having trouble locating Lumiline bulbs and fixtures.
LumilineLives makes and markets LED Lumiline direct replacement bulbs/lamps.
They are priced competitively with what original stock, used or new is still available, they have similar lumens and warmth but, being LED, are longer lasting and far more economical to use.
There is a listing on eBay.
Gene

Here is a link that might be useful: lamps/bulbs

Best supplier of replacement Lumiline bulbs is LumilineLives.com or LumilineLives@yahoo.com

https://twitter.com/lumilineage/status/1248254148828639233?s=20