I agree with Freedom. It is up to you when you get another kitty. I waited almost a year before I got another one after my Taz died.
Bruce wanted another dog about a week after he lost his Cocker 6 years ago. We got Zoee three weeks after DJ passed.

If you could find an young adult kitty that has experience with dogs that would be perfect. My shelter sometimes knows if cats are dog friendly from foster homes or information given when/if surrendered. But a lot of times we don't know.

I also agree with chocolatepuppy that supervision is a MUST. Especially at first.