I know it's not a popular choice but I would have gone with 'Arry. He's a bit of a joke these days and clueless tactically, but he has built good sides in the past by being a good man manager. He clearly knows what a good footballer looks like too. Look at what he did for Lampard, look at what he did for Bale who was stuck playing leftback when he took over Spurs.
Redknapp strikes me as someone who'd be good at working within the FA with long-term administrative stuff too without being on the training pitch every day. And he's a solid presence on the sideline. I reckon it's exactly what the players need. More than anything else they need a motivator. They haven't had that since the early Eriksson years.