"desperately keen" is a correct and usable phrase in written English.
It is typically used to describe someone who is very eager or enthusiastic about something. Example: Sarah was desperately keen to join the school's drama club, as acting had always been her passion.
EchoStar, America's second-biggest satellite-television operator, is also desperately keen to gain control of Hughes.
Twice as sane and half as appealing was Labour, where we synthesised surprise at the suggestion that Cherie wasn't desperately keen on Gordon, and presumably would have gone into anaphylactic shock had it emerged she likes a bargain.
Smaller operations are often uncovered only when the electric lights start fires, or when local teenagers mount a burglary.The police and the courts can neither keep up with the surge in small-scale production, nor are they desperately keen to do so.
He said the board were "desperately keen to keep him in New Zealand" and were "certainly not going to leave him without options".
If you find yourself in front of young, inexperienced tutors and lab leaders this term, they're probably a bit nervous and desperately keen for you to ask intelligent questions.
He is desperately keen for fame, even infamy," Wilson says.
He named several seats where his party could win at Westminster level, and claimed the press was "desperately scared" on UKIP, and keen to run stories attacking his party.
Executive producer David Greenwalt was very keen to acquire the character of Cordelia for the spin-off series, commenting, "I desperately wanted her to come to Angel because Angel being dark and broody, we need a big bright smile".
Harlequins, keen to bring their limping fly-half, Nick Evans, back into the fray towards the end of a desperately tight match, said they had replaced the former All Black early in the second half for tactical rather than injury-related reasons.
Carolyn will be remembered by those who miss her desperately everyday for her keen sense of humor and her artistic eye.
