Installing Extractor fan ducting to outside wall through small cavity.

My kitchen has a high humidity (averages the 80's and frequently 99% humidity on my Digital Thermo-Hygrometer) I do open the kitchen window... thats when humidity sinks to the 80's. Wren kitchens installed my kitchen a while back. In the process we discovered the back wall (where the oven \ extractor fan is placed) is a false wall. The cavity is less than a foot at the widest part. House was built in 1900 and wall is about a foot thick. I noticed my cupboards along that false wall smell musty\ mouldy and dont smell safe to use. I've also noticed that the hood extractor just recirculates the air in the kitchen rather than actually extracting the steam \ damp to the outside. I'll need to get someone in to fix the extractor so it extracts the damp air outside. What would be my best options to resolve the musty cupboards? I was thinking of adding an extrator fan on a solar panel to extract the air from the cavity behind the false wall. Is this the right way to go? Is there anything else I should be considering? How much should I budget for this work?