I've got such a wonderful secret,would you like to know it as well,I told it to ***** this morning and I'm quite sure ***** won't tell.
Now come on right into the corner...

It's all over the Internet with "writer unknown" or "anonymous."

Here's the poem --

I've got such a wonderful secret,
Would you like to know it as well?
I told it to Pu$$y [my cat] this morning
And I'm quite sure that Pu$$y won't tell.

Last night I crept into the study.
And Nobody knew I was there,
And I found such a beautiful dolly
- in a box - on Dear Daddy's armchair.

Now Daddy's too old for a dolly
And Baby is only just three
And I've got a birthday tomorrow
So I think that dolly's for me

**********
All I can find is Anon. Maybe call your public library's reference desk? They will use Google but also have print resources that might list an author. It seems to be a UK poem from long ago.

Thank you x