spinoff.nasa.gov
It says 1 billion dollar per engine development and then states the difference between development time periods of commercial and military engines. They stated the difference between commercial and military one when there is. So I will assume no difference in price considering no difference stated.
Published in 2004 so 1 billion dollars then is approximately 1.67 billion dollars today. Let's add some technological advancement gap and make it 2.5 billion. This is also the predicted cost for engine of Tempest. The F135 costed 10 billion but it's not a suitable engine to compare because of vertical take-off stuff. Safran and MTU signed for 3.5 billions for FCAS. So let's say it 3 billions as a average cost. I guess this money is already saved by giving the F-16 modernisation program to TAI, even if it is more than 3 billion.
And it's stated as 10 years to develop a military engine to Nasa. Since they have a lot of know-how and we are rookie, I'll add 5 more years. And let's say real designation process started at 2020 but not in 2017 (when TRMotor established). It still towards us to 2035 not 2040. And you stated "Kaan will not be ready in 10 years even with F110". Kaan likely to be ready with TF-35K around 2035 at latest. And much earlier with F110. According to these data and some guesses of mine.