Dawes Discovery upgrade

Hi everyone I have got myself a silver Dawes Discovery 701 second-hand after not having a bike for 20 years. I love it it's so nice and light and feels great to ride but as it was second hand I need to upgrade some parts and would appreciate some advice on getting parts. I want to put a new 9-speed freewheel on the back as the parts on it now are not matched. It has a Shimano 105 rear derailer and a Shimano 9 speed shifter but.. its back wheel has a 5 speed Shimano freewheel on a Pelissier Hub. Learning all about these parts was mind-boggling, to say the least, but all I really want to know is can I get the 9-speed freewheel on the wheel and a new chain as everything else seems ok. Here are some pics if that helps