I stayed at the Çatlak Hotel
when I walked the Ihlara valley in June 2 years ago. When I left I visited Aksaray museum for a while, so it wasn't until about midday that I actually departed from Aksaray. I did a bit of sightseeing along the way - Ereğli and Can Hasan - and I made it to Side at nightfall - about 8.5 hours after leaving Aksaray.
Antalya is about another hour from Side, so I reckon you should have no problems doing that drive and stopping for sightseeing along the way, just depart early.
P.S. I see that Lystra is only 10km off the main road from Konya to Seydişehir, not much of a detour.