I went to turn on the desk lamp that is the night light for the kitchen. The Westinghouse F15T12D just glowed dimly. I tried 3 more  times same result with no starter working. I took the bottom plate loose and the starter socket fell apart. GRRR! I don't have any spares, so off to eBay it is. No place around here has any and that includes the hardware store that has antique stuff. I had another desk lamp that I had converted to a starter start, and it has a missing wire nut, and I couldn't find those either. They are in a "safe" place....wherever that is.  I finally got another desk lamp that accepts T12 lamps and put in the Westinghouse. 

Mordern starter socket does not fit in ?

Can't FIND modern starter sockets . I can get lamp sockets with starter built in. Can't use that. Remember the US is not preheat / switch start unless it is 20 watts and under, and modern 20 watt units have been trigger/rapid start for some time.  I think eBay it will be.....   

Would any of these starter sockets work for your purposes?

http://semperlite.com/products/lampholders/list.php?B1=61

Yes, and I found some on eBay. I will be ordering later this week.

 Starter sockets came, got wire nuts and repaired unit. RATS! My F15T12D Westinghouse lamp (well I bought a case of these) already has a little black spot on one end. I put a Dura FS-2NA starter in. That's an automatic reset starter as the socket is in the base of the lamp with the ballast. I also repaired another lamp with a wire nut as I had converted this one to starter start and not a button. Whoever had it before had put in 2 switches one for a manual preheat. I had to replace the ballast with that one as the bulb was VERY bright and after 30 minutes the ballast was so hot you couldn't touch it.

Is the lamp blink happy ? If it is this may be the reason for the end blackening

And what type of ballast is the one that overpowered the lamps/overheated ? Is it of the correct ratings ?

No, the lamp blinked once.  I now have one of the ultra cheap Chinese F15T8D Sunlite bulbs in it I commented on elsewhere. It starts with one blink... sometimes 2.