There is increasing evidence that aerosol transmission is a major contributor to the spread of influenza. Despite this, virtually all studies assessing the dynamics and control of influenza assume that it is transmitted solely through direct contact and large droplets, requiring close physical proximity. Here, we use wireless sensors to measure simultaneously both the location and close proximity contacts in the population of a US high school. This dataset, highly resolved in space and time, allows us to model both droplet and aerosol transmission either in isolation or in combination. In particular, it allows us to computationally quantify the potential effectiveness of overlooked mitigation strategies such as improved ventilation that are available in the case of aerosol transmission. Our model suggests that recommendation-abiding ventilation could be as effective in mitigating outbreaks as vaccinating approximately half of the population. In simulations using empirical transmission levels observed in households, we find that bringing ventilation to recommended levels had the same mitigating effect as a vaccination coverage of 50% to 60%. Ventilation is an easy-to-implement strategy that has the potential to support vaccination efforts for effective control of influenza spread.