I didn't really have much hope but thought that just for fun I would try to find any evidence of such a record on the internet. I googled "little match girl danny kaye" and the very first hit led me to a web page with a long list of old children's story records including the one I was looking for. The record containing HCA's stories narrated by Danny Kaye did include TLMG plus a few others that seemed to ignite recollection in my head so I'm sure it was the one I had heard long ago, and here it is.

The page also includes downloadable mp3 files of the records so of course I did just that and upon listening to Danny Kaye tell the story of The Little match Girl I am now sure that it was the very same recording that left such a big impression on me all those years ago.
If you would like to hear it for yourself you can listen to it right here.
I must admit that now I prefer the silent animated film version I included in my previous post but Danny Kaye had the advantage of being the first to tell me the story and that, when I was still so young. He must have made an quite an impression, for me to search out the Little Match Seller once more all these years later.
Anyway if you or your kids might be interested in listening to some old stories told the old fashioned way, here is the link to the web page.
http://www.artsreformation.com/records/
(The record containing The Little Match Girl that I downloaded is "GLP 74")
Finally, to close, may I just quote another great old story teller and say, "now you know...(dramatic pause)...the rest of the story."
(Um, yeah, that's Paul Harvey)