McKenna, who oversaw three promotions in four years, announced last night that he had quit in order to take a break from management.
Ashton is now on the hunt for someone to lead the team back into the Premier League. They will become only the 20th permanent boss in the club's 90-year professional history.
"On the one hand you have to deal with the emotional distress of a friend and a colleague, a very successful one, leaving the football club," said Ashton, speaking to TownTV.
"But on the right hand we have to move forward and we have to move forward at pace. Let's be really clear, that will happen at pace. We have a season to prepare for, we need to appoint a new manager and we need to move into the player market. So we need to move swiftly."
Strasbourg manager Gary O'Neil is the current bookies' favourite for the role, with several other bosses with Premier League experience likely to be in the mix.
Asked if there was a timescale he had in mind, Ashton replied: "Imminent. But we will be thorough. The process will probably be similar to the process we ran when we recruited and appointed Kieran. It will be very detailed, very thorough, but we do need to move swiftly. Fans need to be assured that we're on top of this and we will make the right decision for this football club as soon as we can."
Reflecting on McKenna's incredible four-and-a-half years in charge, Ashton said: "There are numerous amazing memories - the promotion days in the sun, the (open top bus parades on) Bank Holidays, the town lighting up, some amazing away days, whether that's in League One at Barnsley, the opening day at Sunderland in the Championship, the one at Tottenham (in the Premier League) - they are all special.
"For me, the personal memories are our matchday minus one Costa coffees in his office where we put the world to rights. We don't just talk about football, we talk about everything. Personally, the fact that he travelled the length of the country and stood by my side at my dad's funeral is something that I'll never forget. On and off the pitch I'll forever be grateful for Kieran and his time here.
"I thoroughly enjoyed his company. We've had some real debates, faced some real challenges together, some of which people don't see and will never hear about, but we've also had some amazing successes. Three amazing days in five years here at Portman Road is something to behold.
"Whilst there is sadness, a wise man once said to me 'don't be sad, celebrate the fact that it happened'. I think we need to celebrate the fact that Kieran McKenna came to us as an unknown manager, untested, unproven, and it's been a pleasure and delight to see him grow into a Premier League manager. He's earned the right to move on in the way he wants to move on, but this club now has to move forward.
"I've said this time and time again. We are so much more effective, so much more powerful, when we move together as one. Whilst people will be upset, worried and nervous, we'll do the right thing by the football club and make the best appointment we can. If we move together as one we'll attack the Premier League.
"On the pitch Kieran leaves us in a good place. There'll be work to do this summer, but the new training ground will be open in the next four to six weeks, our academy is in a great place, the stadium is totally redeveloped and the future is extremely bright. We are in a really, really good position."
