lori,
Strangers are welcome everywhere, so don't worry for that.
Older woman? You are not old, really. Here you'll find people of all ages at night clubs.
Alone? Well, lots of men will approach you. Please, don't laugh on their English.
But, believe me you'll go home much later than midnight, never underestimate Madrid night life. You'll have great fun for sure. :jump:

Near the Plaza Mayor there are hundreds of bars (we call them pubs or bares de copas) wich are usually crowded every night. I cannot recommend any particular one. Just go from bar to bar until you find the one you like.
There are some night clubs (discotecas or discos in Spanish), in Calle Arenal you have "Joy Eslava" and "Palacio de Gaviria".
Here you have a list with night clubs:

Lanetro,a leisure time web page