I looked it up*

https://blog.kagi.com/status-update-first-three-months#kagisearch

It's $.0125, so 1.25 cents not double digits like I thought. They also average 27 searches per day per user. So an average of 821.25 searches per user per month, meaning a cost of $10.27 per month.

It's probably as good as we are going to get.

The best options would be an open source, donation supported search engine, but the money required to host/develop that is immense.

We are all freeloading off of Lemmy right now, unless you are donating to the people who are running the servers. The cost to run a search engine is much higher though -- kagi pays (iirc) double digit cents for each search, even before development costs, with the average user doing 700 searches a month. The costs are way higher.