if you are using custom search web part you;re right, but you can use Kentico API to log that data.
Add this code to your method in your custom search web parts before Searching takes place
CMS.WebAnalytics.AnalyticsHelper.LogOnSiteSearchKeywords(SiteContext.CurrentSiteName, DocumentContext.CurrentAliasPath,CultureHelper.PreferredUICultureCode, searchKeword, 0, 1);
it should fill up on-site report.