Hungary/Slovakia trip planning

Trip at end of May, will be there 12 days total, so 5 or 6 days in each country. First time in either country so looking for recommendations of what to see and do.

Mid 50's, me and wife, travel out of country 2 or 3 weeks every year. like history, culture, food, scenery of a country. Really like trying to do the local things. Can't really do hiking, cycling, etc... wife broke both ankles this year and we were already old and fat!

Thanks for any recommendations.