To add to Steven Lee's answer, the entire transaction level dataset from England's Land Registry can be downloaded from the Land Registry site.
This dataset contains every house price from house sales since 1995 and contains both the full postcode, the street name and the street number. If you can handle the entire dataset (~20 million rows) it is not immensely hard to aggregate to LSOA (at least approximately even though postcodes don't map neatly onto LSOAs). An exact mapping might be possible with a little GIS wizardry but the approximate mapping is available using the postcode to LSOA lookups from The census service here. The tables provide lookups to most higher-level geographies from full postcodes.
The datasets are a little large to handle in Excel or Access (there are >2 million postcodes never mind the 20m house prices) so it is worth looking for better tools to process them. My personal choice is Google's BigQuery which is designed for large analytical workloads but is cheap (and for small workloads, free) and available to anyone with a web browser and a Google account.