The pump needle broke off, leaving the base in the nozzle. How to remove the stuck base?

I just had this same thing happen last week. I straightened out a paper clip and bent a small hook in the end. Put the hook through the small hole in the center of the needle base and pull up. It came out pretty easily. Make sure you release the tension on the pump head first.

The "offcut" in the valve head is useless - so there's no point in trying to save it. Your goal is to save the pump head.

Very first thing would be to make sure the lever is in the "loose" position. It may be locked when vertical, or when parallel to the hose. You need to remember which way is loose.

If you have access to left-hand drill bits or an easy-out then simply backing one of them into the valve hole might be enough to pop it free. Another way to get purchase is to use a torx head driver of a suitable size, ranging from a t10 to a t20, and hammer it into the hole. The lands on the torx driver can cut in to get extra grip. It may destroy the torx bit.

You might choose to put some liquid penetrating oil into the head to loosen the threads, but this risks damaging the rubber components, like seals and gaskets and possibly the hose as well. Perhaps this is a later option not a first line of attack. Daniel's suggestion of liquid soap is great advice here too.

Your last resort is to replace the pump's head. They can often be bought as upgrades from a reputable bike shop.