Is there a wifi (LAN) speed tester?

The short answer is no - LAN speed cannot be tested using just a phone itself and needs a proper testing setup to measure it correctly - which most users won't do so it's questionable whether it's worth my time to implement a feature which most people won't use.

Long answer:

To perform the speed test, there must be some device from which you perform the download and to which you perform the upload. The download/upload testing files have to be down/up-loaded from/to some other "server" device on your LAN and the speed of such a transfer will be measured. This requires some setup and ordinary smartphone users want something where you just click a button and it works.

The next problem is the test will be limited by the slowest device you have in your test setup - it could be limited by the computer acting as the server, by the wifi technology used by your router or your phone, by the CPU speed of your phone etc. The server device should be connected to the router using wire otherwise both the sending and receiving devices use the same wifi network which affects the resulting speed considerably (the result won't be just one half of the maximum speed but much less - there will be collisions that will slow down the speed a lot).

If you are willing to invest some time to properly setup your network for the test and know what you are doing, you can get a testing server called iPerf (kind of standard for LAN speed testing) here

and use some iPerf app from the App Store (search for the word "iPerf", there are several of them - I haven't tried them so I have no personal recommendation).