I don't see a place to buy Topkapi tickets on the official website, either. https://www.millisaraylar.gov.tr/Lokasyon/2/Topkapi-Palace. You can try again, at other hours of the day, to see whether that changes things.
Like Simon, I just showed up and bought my ticket at the site. There was a bit of a line, but nothing really significant. And I know I didn't show up super-early.
For me there wasn't much to see in Hagia Irene. The Palace is huge and can take a full day. It's too bad the tickets are good for just one day, so you have to cram everything into that one day. However, it would be a real administrative challenge to extend the validity and be sure the tickets were not shared.