As long as you have a valid marriage license, you still can file as Married, Filing Jointly. Everybody has an obligation to pay taxes, whether they are here legally or not. Your wife should obtain an Individual Taxpayer Identification Number (ITIN). See this link for information on where to go to apply for one.
https://www.irs.gov/individuals/international-taxpayers/acceptance-agent-program
I got my information from this source. You should have a look at it. Good luck!
https://www.sapling.com/10001765/tax-laws-marrying-illegal-immigrant
