How to connect an ethernet printer to WiFi?

I need an adapter that connects to the existing WLAN and offers an Ethernet connector for the printer.

Use a Wireless bridge:

A wireless bridge adapter, like the Asus WL330GE, simply "catches" the Wi-Fi signal and converts it to Ethernet, letting connect those Ethernet-only devices without a long cable run.

Just note that these products often involve some elaborate setup routines (such as installing software on a nearby PC, or accessing a setup screen through a web browser) that skew towards the skillset of an advanced user.

Source How to get your Ethernet-only gadgets on your home network - CNET

That particular device is expensive at $189.95 and probably overkill for your needs but there are cheaper/simpler devices available:

Laptone N300 Wireless Mini Router

Laptone Wireless Mini Router is designed to conveniently extend the coverage and improve the signal strength of an existing wireless network. With 300Mbps wireless N speeds, Range Extender button, miniature size and wall-mounted design, extending a wireless network has never been easier.

What's more, its Ethernet port allows the device to act as a wireless adapter to turn a wired device into a wireless one.

Typical price £14.99.