Cool looking project. Ive always liked those old lights - so much nicer than anything you can get now.
I have built a number of all sorts of lights over the years using may different LEDs. what you want to do is pretty simple. If it were me, i would use a small 12v transformer to replace the ballast, then just use 12v LED COB tape to replace the light. I would also replace the vintage internal wire, and to make it easy on you id get the COB specific connectors, which can be bought is small assortment kits - no soldering required. All these parts are easily available on amazon. For that size light the smallest stand alone transformers are enough (i cant imagine you exceeding 5w) COB LEDs i prefer because the light is more 'even' and they can be cut more frequently to get the length right. With some patience i bet you could get the led strips inside an old T5 bulb if you really want the vintage look
a few tips: LEDs come in different colors. white is the only reasonable option here, and i use 4500k lights all the time. They are very 'white' but not blue good to work under. You may want color closer to 2500k-3000k for a more traditional yellowish light.
buy the highest density COB tape you can. More lumens per foot is your friend for a task light. Not bright enough is a bummer. it cost a bit more, but is very worth it.
.... which leads to adding a dimmer. If you can i'd add a dimmer. That means making sure your transformer/driver is dimmable (a bit more $) i like to use the capacitive dimmers for leds, which i dissassemble and rebuild into custom switches using small brass rod touch points soldered to the orginal capacitive sensor. ive been able to add about 10" of 22ga wire and still have them work. its a cool trick. or you can just use a switch that replaces the one you have, but is a dimmer.
a lot of the lights i have made were actually constructed from cheap LED desk lamps, completely dissassembled, then rebuilt into interesting wood/stone custom fixtures. i found that was helpful in learning how the whole things works. you get a 120v plug, then a transformer, dimmer, and often a usb plug for charging - which are designed to work together. all features which are nice to have in, suprise suprise, a desk lamp. might be worth a $20 lamp. if you go that route, and want to use the parts in your lights, make sure the LED output is 12v. COB tape needs that voltage and some lamps are 5v output.
I have built a number of all sorts of lights over the years using may different LEDs. what you want to do is pretty simple. If it were me, i would use a small 12v transformer to replace the ballast, then just use 12v LED COB tape to replace the light. I would also replace the vintage internal wire, and to make it easy on you id get the COB specific connectors, which can be bought is small assortment kits - no soldering required. All these parts are easily available on amazon. For that size light the smallest stand alone transformers are enough (i cant imagine you exceeding 5w) COB LEDs i prefer because the light is more 'even' and they can be cut more frequently to get the length right. With some patience i bet you could get the led strips inside an old T5 bulb if you really want the vintage look
a few tips: LEDs come in different colors. white is the only reasonable option here, and i use 4500k lights all the time. They are very 'white' but not blue good to work under. You may want color closer to 2500k-3000k for a more traditional yellowish light.
buy the highest density COB tape you can. More lumens per foot is your friend for a task light. Not bright enough is a bummer. it cost a bit more, but is very worth it.
.... which leads to adding a dimmer. If you can i'd add a dimmer. That means making sure your transformer/driver is dimmable (a bit more $) i like to use the capacitive dimmers for leds, which i dissassemble and rebuild into custom switches using small brass rod touch points soldered to the orginal capacitive sensor. ive been able to add about 10" of 22ga wire and still have them work. its a cool trick. or you can just use a switch that replaces the one you have, but is a dimmer.
a lot of the lights i have made were actually constructed from cheap LED desk lamps, completely dissassembled, then rebuilt into interesting wood/stone custom fixtures. i found that was helpful in learning how the whole things works. you get a 120v plug, then a transformer, dimmer, and often a usb plug for charging - which are designed to work together. all features which are nice to have in, suprise suprise, a desk lamp. might be worth a $20 lamp. if you go that route, and want to use the parts in your lights, make sure the LED output is 12v. COB tape needs that voltage and some lamps are 5v output.