My mother is an illegal immigrant and thus I cannot place a SSN for that section. Is there anyway to get around that or would I have to file by mail because of this?

without a social security number or ITIN you can not claim her as a dependent regardless of whether you e-file or file by mail.  if you list her then Turbotax won't even allow you to submit the return.  and if you file by mail and list her without an id number the IRS will change your return or send you a notice - you will not qualify for head of household status.