The answer turned out to be simply that the database had not been started as a Windows Service. This was because the Log On account it was trying to use did not have appropriate permissions. Not having another machine to check the correct settings, I decided that in the Services snap-in (Start -> Run -> services.msc -> SQL Server (MSSQLSERVER) & SQL Server (SQLEXPRESS)) I went to the Log On tab, selected "Local System account", hit "Apply" and then "Start" under the General tab.
For those two services, on my machine at least, this got everything working again. I was only able to discover this solution after googling and ending up here:
In case the post with the answer disappears, here it is in full: