I can't tell you much about Elizabeth Keller. What little I know is derived from census records, and some brief biographical notes dealing with the children (see, for example, the sketch of Isaac Ax available at http://www.heritagepursuit.com/Tuscarawas/TuscarawasBWayne.h... ).

Summarizing the census data:

In 1850, she is listed in William Ax's household, but still under the name of Elizabeth Keller. Age is given as 22, birthplace is Ohio.

In 1860, her name is given as Elizabeth Ax,. Her age is given as 31, and birthplace is again Ohio.

By 1870, she seems to have disappeared. Her children, Mary A., and WIlliam Eli Ax, both appear in the Daniel Zigler household in Franklin Township. Mary, age 14, is described as a "domestic servant", but her brother is "at home".

I have neven been able to locate any marriage record for William Ax and Elizabeth Keller. This may have been a common-law marriage, but I don't know for sure.

That's about all I know. So she was apparently born somewhere in Ohio around 1829, and she may have died in the 1860s.