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Each case is very different and this "sounds" like an open and shut case in terms of liability (it's been admitted).
But a quick settlement isn't always the best thing, as both sides need to fully understand the nature of the injuries and more importantly the nature of the recovery and often this require months rather than weeks of assessment especially where a head injury is concerned.
Generally yes whoever your solicitors are your going to need to chase and chase.
In my experience they take time and you'll need to stay on top of them to get your case resolved. I was also hit with the driver claiming all responsibility with it taking over two years to get the case sorted.