I guess I gotta ask the question. Are you willing to
trust your life to a used tire (that I assume someone else drove around on for god knows how many miles) in order to save a few dollars? Tires are about the last place you want to try and cheap it out, even if you don't have much money; but its up to you to decide on that issue.
While you can replace just one tire its not recommened. The recommended suggestion I've always been given by tire guys is to replace both tires (back or front) at the same time. Otherwise you end up (so they told me) with tires that have different mileage on them so one or more will wear out faster than the others causing possible loss of traction, tires out of alignment, and other issues.
One thing you didn't mention is what happened to the original tire before you put the donut tire on? If you still have that tire and it was a simple air leak or puncture and its on the tread you can usually have a local garage or tire center like Sears fix the flat and remount on the car for a small fee.
You haven't posted your Corolla year and model. Posting that, along with how many miles are currently on the tires may help for others to provide more help. You can find the OEM tire size at
http://www.tirerack.com. Go through the Update Garage to select your year, model, etc., and then select tires. Then you can view the Original Equipment Size Tire for your car. For example for my 2005 LE the OEM sized tire is 195/65-15 (195/65R15). With the correct tire size you can check around locally and on websites like craigslist.org for new/used tires in that size, in your area.
Please post back on how you make out so others can learn from your experience.