Speaking at the first meeting of the full council since the August recess, Coun Box said he saw "no reason" why he shouldn't carry on in the role he has had since 1998.
A 'no confidence' motion in his leadership was signed by 23 of Labour's 52 district councillors before a party meeting on Monday, but it was narrowly defeated.
Asked directly by Conservative opposition leader Nadeem Ahmed whether or not he would still be in his post in May 2019, Coun Box replied: "Yes, I think so. I see no reason why not.
He later restated his desire to continue, saying: "Will I still be here in May 2019? Yes, I will be."
The meeting saw another fierce debate about the state of Wakefield's children's services, which were deemed to be failing vulnerable children by Ofsted in July.
Tory councillor Gill Cruise, who received well wishes from the chamber after undergoing cancer surgery last week, branded the findings "disgraceful".
She said: "When I was at school and you weren't doing well, you were put on report. Who is going to be put on report for this?
"Who is going to take responsibility? It's completely unacceptable."
She said it was wrong to blame austerity for the problems, adding: "Councils up and down the country are going through the same cuts, and their children's services aren't in dire straits."
Labour member Steve Tulley, a critic of Coun Box, said he had "grave concerns" about children's services, but claimed a Conservative-led council would make the situation worse.
He said: "I'm struggling with the leader of the council, but that's for another place.
"I'm losing confidence whether we'll be able to get out of it (an inadequate Ofsted rating)."
Children and young people portfolio holder Margaret Isherwood said every council member had to "own responsibility" for making the service better.
She said: "We have always accepted the situation. That's why we've put the improvement plan in place.
"We've never tried to deny that there are problems. The problems are being attended to."