So, you have got your engineers report back and it says everything is above board. Well done! You are well on the way. Maybe it brought up a few small issues - if so you can consider going back to the vendor (seller) and seeing if you can wrangle a better deal.
Next step, when you have informed your mortgage broker or building society is for a valuation to be done. This costs an average of €150 to be done and they will have a list of valuers available (they will generally select one themselves). These can be auctioneers in the area, or engineers, so do not be surprised if it is someone you have already had dealings with. Prices actually vary here, so if it is possible you can try to get a few quotes. We paid €130, but got a different quote of €180 so there is definitely a bit of difference in the prices you can get.
There is very little for you to do in this step. Provide the mortgage broker with the selling auctioneers details (or if it is a person selling directly their details) and they will sort out a time to do it. Generally this takes up to a week. This is very important as it tells the mortgage people that the house is worth what you have said it is - making sure their money is safe!
When this is done, the valuer will send the valuation to your mortgage people and they will let you know. At that stage they will start the process of moving your approval in principle to be a formal offer.