Will a suitcase with a broken handle be accepted as checked in luggage?

It is unlikely they can accept that. It may even exceed the dimensions allowed in which is it is pretty much an automatic not acceptable. Otherwise some airlines may let it in. The last time that I checked in a cracked suitcase that was held by ducktape, the airline put a fragile sticker on it and another one that I had to sign a waiver saying some like Improperly packed contents, airline not responsible.

Your best bet is to remove the handle mechanism. Try to unscrew parts until it separates completely. Sometimes putting it back together after will make it slide back. Even if they do let your check it in, it is unlikely that the suitcase will come out intact. Even good condition and really sturdy suitcase get damaged and its usually the wheels or handles that break first.

It'll depend on the airline. Most airlines have baggage dimension limits, even for checked luggage. You'll need to look on their website.

Alternatively, you could try to fix it yourself if you have time and some tools, or worse case, take out the screws from the inside when you get to the airport, so that you can actually remove the handle, and place it in your bag.

Then when you get to the destination, reattach it and carry on as you were.