Patience alone might not be enough. I have seen planets that held out deep within my zone of influence forever - the "influence battle" depends on many factors like the population, planetary attributes, planetary improvements, the techs of the race etc.
Look at the tooltips of the planets and the surrounding tiles - if you do not have an influence many times higher than the owner of the planet, it will never flip. To increase your influence, you need to develop your nearby worlds in terms of influence - more population, cultural improvements, cultural starbases ...
Also you need to look for the enemy planets with the highest influence and try buying it. I have seen planets that had such a high influence they switched the zone of influence for hundreds of tiles when I bought them. While other planets with low influence might not even give their own tile as zone of influence.