Is there an app available from the App Store (iOS) to find out the LAN speed? I'm trying to see if it would be worthwhile to upgrade my PC computer to a 5 GHz wireless adapter. The only device I own that is capable of using the 5 GHz band is my iPad. So, I'm trying to see if it's possible to measure LAN speeds on my iPad. So, any ideas?
If you want to measure the speed (bandwidth) between your iPad and your PC, take a look at iperf.  It is available for computers and there are versions on the App Store for iOS too (the one I use seems to have gone away, but searching the App Store for "iperf" reveals several possible options).

You run an iperf server at one end (eg the PC), and a client at the other (eg the iPad).  It runs for a configurable amount of time (I think the default is 1 minute), and then tells you how much data it managed to transfer.

Compared to eg copying a file, the figures you get this way are a more accurate reflection of the available network speed, because copying a file requires retrieving the data from a hard drive, writing it at the other end, sending acknowledgements, the overhead of the file transfer protocol etc... iPerf is simpler than that: it just transfers data as fast as it can.