Obviously build more: we need to densify our cities and their inner suburbs, and make it easier to build houses than it currently is. I'd move to a deemed consent model where if a planning application fits the local authority's development plan and rules, it has to be built: no objections from the community. We need to get better at building infrastructure, particularly reservoirs, to let us do the housebuilding in the areas where it's required. The market is pretty good at adjusting house prices based on supply and demand, so increase supply and prices should fall, and some of the unusually cheap properties being touted above will be cheap due to low demand (for a reason!).
It'd be more marginal in terms of impact, but on the demand side I'd require planning approval to change a property from a primary home to a second property (either empty for the owner to visit occasionally or to rent out as a short-term holiday let): cap the percentage of second properties at say 20-25% of a community and then require primary homes to be either owner-occupied or let out on a long term basis.
Don't necessarily believe that foreign students are a huge driver: I don't need to look very far in Cardiff to see that there are whole new accommodation blocks being built to accommodate foreign students, so increasingly (given that we tend to only get wealthier overseas students given the size of the fees!) they're not part of the demand for traditional houses and flats. Also, we're far from the only country in Europe to have high levels of immigration (student or otherwise) but few other countries seem to struggle to house their populations quite like Britain does.