It is my understanding that the API does not provide support for searching your provided KML:
google maps API geocode finds nearby KML
Google provides a service called placesService ( https://developers.google.com/maps/documentation/javascript/places ) which lets you make requests to their database of content and return results in the form of KML near a location within some bounded area - but to my knowledge you cannot specify the source to search. (You could try testing the placesService, with a KML layer set on your map and see if it includes your data).
Assuming you are generating your KML from a database, can you could regenerate it appropriately filtered or styled based on the user's search terms. This seems the most likely solution without additional details.