Once installed and set up, the Static Search feature usually works just fine.
So you’ve invested in superb ‘Static Search’ plugin from: https://marketplace.getpublii.com/plugins/static-search/ , installed and configured the plugin as you want.
In Publii > Tools & Plugins > Static Search, you’ll have turned that on, clicked the plugin button to open up the ‘Static Search’ settings page, and configured your settings, yes?
To recap, the Static Search plugin is quite powerful so have a range of settings you can configure. The documentation is at: https://getpublii.com/docs/static-search.html
Initial thoughts: maybe double-check your setups.
Which version of Static Search plugin are you using? At the time of writing, the latest is v1.2.0.