To do this and keep the lines, it would probably be best to replace the socket, although you could repair it as well by soldering the wire back in place. What you describe sounds like what has happened to me. The wire has broken, right at the part that is the contact point at the bottom of the bulb. The contact is pressed into a fiber disc(one contact for a single filament bulb, two contacts for a dual filament bulb), and can be pushed out. The repair I did was to bring the wire through the hole, and fashion the wire into the shape of the button and apply solder to keep it's shape.
Then, the trick to the LED conversion is the flasher unit. The stock one relies on current to flash, so if a bulb burns out, it doesn't flash, and you know right away if a bulb is out. It probably wasn't designed for that, but it is handy. The thing with LEDs is that they don't use but a fraction of the current that a bulb uses, so the standard flasher unit won't work. You need a flasher with three wires(the extra wire needs to be a ground). Some bikes already have a ground wire right at the flasher, like they knew you might need it someday. Standard base LEDs are easily found online (
www.superbrightleds.com is the one I remember), and these will fit inside your existing lenses, maintaining your lines.
Good luck.