What is wrong with this sentence, and how should I fix it?

Perhaps

I was fortunate to be named chair of this board shortly after I joined.

Or, if you don't want to emphasize how lucky you were,

I was named chair of this board shortly after I joined.

If it is tense/aspect that you are after, then maybe:

I am fortunate to have been named as chair of this board shortly after I had joined.

The simple past of 'I joined' doesn't stick it in sequence very definitively (the 'shortly after' does of course).

I was fortunate to be named chair of the board shortly after I joined.

It sounds like you have the worm. This sentence has an overly formal sound to it. Try this:

Fortunately, I became chair of the board once I joined.