Attention Getters
The pilots did well to land as they did.
Unfortunately, the communication failed to get across to the controller as there was no 'attention getter' phraseology. Remember it may sound silent on the frequency you are on - but the controller may have a lot of other communications to deal with that the flight crew will be completely unaware of. Therefore an 'attention getter' phrase - like "MAYDAY MAYDAY MAYDAY" shuts everyone up and they all listen really hard -- Then when you have their full attention, callsign so ATC (and others listening) know who has the problem. If you feel that sounds too 'trite' for a US operator who prefers to sound laid back how about "EMERGENCY EMERGENCY".
After the 'attention getting' and callsign, it would be nice if the rest of the message is clear and provides all the information needed but in an emergency there may well be more to think about. The important thing is to shut everyone up and make them take notice of your problem and know what you need NOW. If you aren't going to shut them up and make them listen then you may as well not bother and just assume they'll realize from what you are doing and somehow guess what you want.
In the same way there are attention getting alarms in the cockpit as well as an EICAS - to shut up any chat and get your attention here NOW!! then handle the event. In that order.