Household employee wages not on a W-2 have their own specific place on form 1040, which is line 1b. Hopefully, if you follow the links for "other wages not on a W-2", one of the options will be household employee. (I have not tested it.)
You are correct that you do not pay household employee tax, she does not pay SS or self-employment tax, and she does not get social security work credits.
