From a technical point of view, the honours are awarded by the monarch. The honours list emerges from a consultative process involving the Civil Service and 10 Downing Street. The Civil Service approaches hundreds of interest groups like trade associations and local government associations to ask for nominations. Although as a constitutional monarch, the Queen would not theoretically interfere with the contents of the list, I am sure she suggests people and even crosses some people off!
What I am getting round to is that MP's don't have much say in who gets honours, and an Early Day Motion in the House of Commons would have no effect other than to remind those who are drawing up the honours list that Kenny deserves a knighthood.