Fill one side of your kitchen sink with HOT water. Make it as hot as you can tolerate because it will cool down fast. Usually the temperature out of the hot tap is fine depending on the temperature settings you have on your hot water boiler. Then fill the other side with ice cold water. Dumping a bowl of ice cubes in the cold side will work nicely.
Soak your arms for one minute in one side, then plunge your arms into the other side for one minute. Do this going back and forth about seven times in each basin. Initially the hot or cold may seem unbearable but you will get used to it quickly.
The hot water will promote circulation in your arms and the cold will reduce inflammation. This is not a cure but may help in the recovery process or at least alleviate some of the discomfort you may have.
Circulation is important for healing because the increased blood flow helps to rebuild damaged tissue and will also carry away toxins and damaged cells.
Remember, inflammation is nature's cast. It is your body's way of warning you not to use the injured body part until it can heal. Eliminating the discomfort doesn't resolve the problem. Don't fix symptoms.
I also suggest that you consider washing dishes by hand rather than using a dish washer. The warm soapy water is beneficial and can be a relaxing therapy session for sore hands.