I have been through the very same situation.
There is no easy way, but it may be made easier if your father's wife knows about you.
If she doesn't, then i would back off altogether.
If she does...then you have his name, can get his phone number and then ring asking him if he knew your mother........and go on from there.
In my case, his wife knew about me, they hadn't had any children and i had always been a bone of contention.
He didn't want to know, the telephone conversation lasted less than 5 mins, Mrs sqad cried........but it didn't bother me a jot...and never has done.
That is the only way........forget an intermediary.