How to get the engine to run correctly after the catalytic converters were stolen?

Replace the parts that were stolen to get it to run. Depending on which state you live in you may be able to use aftermarket units and save a lot of money. If they cut the pipe under the car like the first image and didn't get the front two converters that bolt directly to the exhaust manifolds, they only got the rear unit. If that is the case and you live in a state where they don't test or examine the car you could just weld a chunk of pipe in there as there is no monitor for the rear cat.
You also may want to report it to the police and your insurance company. Some have been covering them under the vandalism section if you have it.

If you mean they also cut that entire pipe out, you can buy that pipe as a replacement for around $300.00 or so. Unless you live in NY or CA. In those places you need to use either OEM or CARB approved units and for an 18-year-old car with 165,000 on it, those parts will cost you more than the car is worth. As for any other solution, there really isn't one due to the design of the vehicle. If they cut off those pipes ahead of the pipe shown you would be looking at closer to $2,300.00 if you can use aftermarket parts.