I've not been inside that TV specifically but I'd expect an input board dealing with the HDMIs. It's at least worth opening up and seeing if there is anything obviously wrong, e.g. checking ribbon cables between the boards, checking for dried caps.
I would also suggest having a serious Google hunt based on different versions of the model number, and you may come up with the service manual or replacement boards / parts for it.
If you need to take it to a repair shop to do the work around here the cheapest guy is a minimum $200.
Take that into account on thinking about a repair shop.
But I did a quick search on ebay for "Bravia parts" and plenty came up.
the HDMI and analogue signal parts are definately separate boards. if you are adventurous enought search for your exact model.
though the hdmi boards I saw seem to be about $120.