Your mother-in-law needs to apply for a visitor visa from the nearest US consulate. Click the following link to search for a consulate: http://www.usembassy.gov/
Although not completely clear to me, it sounds like you step-mother wishes to come to the U.S. to visit for a short time and then return abroad. Generally, applicants for such a visit need to apply for a visitor visa (a visa in the B1/B2 nonimmigrant visa category). That type of application is handled through the U.S. consulate's office in the foreign national's country, and part of the process includes providing documentary evidence to show a genuine intention to visit temporarily and then return home. Typically that type of documentation includes evidence of one's ties to her country, such as evidence of a temporary leave of absence from work, evidence of continuing to have a home abroad, evidence of financial ties abroad, evidence of other family members remaining abroad, etc.
There is not enough information to answer your question. What you seem to be referring to is a letter of invitation which is used for some countries but not others. Depending on which country she is coming from, she may or may not need a visa. Timing will also depend on which country she is coming from. There is never any guarantee that a non-citizen will be admitted into the United States. Good luck!
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