Kelly had previously worked at TES’ parent company, TSL Education, for 22 years and was also editor of Times Higher Education (THE).
He will be joined by Mark Cooper, who earlier this month sold his boutique PR agency Van Communications, which he started in 2004, to Richmond Towers Communications.
The pair previously worked together on the relaunch of TES in 2011, which went on to win 'Business Magazine of the Year' at the PPA Awards.
The third partner joining the start-up is Andrew Moss, a City-trained lawyer and former broadcast journalist.
The consultancy - which is based in Swiss Cottage - will help organisations and businesses in the education sector that want to use education to leverage their brand.
GKP will specialise in helping schools, universities and corporates, but will also advise unions and regulatory bodies in the sector.
GKP founding clients include Ofsted, National Association of Head Teachers, Keystone Tutors, YouthSight and researchED.
Kelly said: "Education is full of fine initiatives and potential stories, but not enough of them are told well enough to make them memorable. The GKP team are not educators. We are communications experts who know a lot about education. We will help clients explain to both target audiences and stakeholders why what they do is important and how what they do is different."
Cooper added: "Our aim is to be an inventive, insightful, candid and results-focused agency. Our knowledge of the education sector is second to none, our owned and earned media contacts are excellent, and we’ll never be afraid to challenge our clients to think bigger and go further."
The consultancy has already been appointed to develop a global teaching and classroom kit in support of the United Nations International Year of Pulses and is providing crisis consultancy for an unnamed client in the education space.