Too few details here, but you should contact a lawyer.
Sounds like maybe there were two different agreements here:
1) The email or whatever the arrangement was that enabled you to work the last two days; and
2) The contract saying you'll start on Jan 17.
Generally, contracts are enforceable as of the date that they're signed... not the date that they "start". It would depend on what state you're in a bit, but most employees (even under a contract) are "at will". Meaning they can terminate you at any time. But if your contract (either the Nov email or the later "contract") specify that they are hiring you for a certain number of days: (for example, "You agree to work for us from January 2-19 from 9-5 daily), then that may be enforceable. Were you in the "trial period" (however that was defined?)
You might be entitled to some severance, but generally there is a duty to "mitigate" damages. meaning, they may be on the hook for some portion of your time, because you relied on their offer of employment. But you are not entitled to sit on your butt and eat bon bons for the next 60 days on their dime. YOu will have some obligation to start job hunting. If you can't get another job until 30 days from now, they may have to pay the difference.