The pacing of the chapter did seem a bit fast but I do get the idea of doing an immediate execution. While there may be some benefits to doing a deeper investigation, Laylin already seems to clear on the corruption, it'd be more of a thorough verification of what she already knows. Also his traitorous statement alone (depending on ruler's temperament), not to mention the attempt on Del's life, would be enough to warrant execution. Having a the execution immediately in front of all the gathered nobles helps to send a clear message that any treason will be swiftly dealt with, even if its under the pretense of helping Sophen. As there might be others who may hold some thoughts on rebelling, this would serve to quell that. Moreover, this would also show that Del and Laylin are united and that there is no use in trying to appeal to Laylin alone. Though it may seem a bit gruesome by modern standards, this isn't unusual treatment for traitors, especially in the circumstance where you are establishing a new rulership.