Kumar, who has led the channel since its trial stages, will move to another "senior role" in the corporation, to be announced shortly.
Nagler's priorities are to "focus on growing the channel by driving consumer take-up, working with in-house and independent producers to increase HD programming, and leading preparations for BBC HD's launch on Freeview", said a statement. Since joining the BBC in 1996 as a journalism trainee, she has worked on a variety of projects including Charter renewal and supporting Mark Thompson.
Nagler will report to Simon Nelson, controller of multiplatform and portfolio, who said: "HD is a central part of our broadcasting future with huge potential to grow on cable, satellite and in the future on Freeview.
"I'm confident that in Danielle's hands, the channel will make the transition to become a mainstream choice for our audiences."
Nagler said: "At the moment there are 10m HD-ready sets in the UK but only a minority are actually used for HD viewing. It's clear there's lots of work to do to open up the enormous viewing potential HD can offer for audiences.
"BBC HD tries to offer viewers the best of the BBC's HD programmes, so I've got the fantastic job of cherry-picking from all the BBC channels to create the best possible showcase for HD content.
"It's a critical time for the channel and the technology, and for people making programmes. But I hope that the BBC can help both audiences and the wider industry make the transition to HD."
Jana Bennett, director of BBC Vision, said: "Seetha's energy and commitment have been vital in taking BBC HD through its trial period, launch and then setting it on course for a successful future. I look forward to working with her in the future."