Hmmm, I'll try to borrow a hub from someone ... or maybe I could just connect the arduino to my laptop's ethernet port and bridge the connection over to my wifi ...
Another funny thing I experienced earlier today:
//client.println("GET /tools/lcd.php?key=1257 HTTP/1.0");
client.println("HTTP/1.0");
This actually led to a "SUCCESS" on my lcd screen - which means (as far as I can tell): There's something wrong with the GET-request (and/or the server answering the request). The downside: I cannot reproduce this little success.
What I'm going to try now: Set up another webserver and see if I'm able to connect to that one.