Help me parse this sentence so I can understand what joke my ancestors played on the King

Solution 1:

I don't think there is any joke at all (or I might have missed it altogether).

Requesting that the most important notables of a city would turn themselves in to the mercy of the King in linen and with a noose around their neck is not unheard of.

The most famous examples actually comes from the reddition of Calais in 1387 to Edward III. This is the subject of a famous sculpture by Rodin of which there are several versions.

After a long siege...

...Edward offered to spare the people of the city if any six of its top leaders would surrender themselves to him, presumably to be executed. Edward demanded that they walk out almost naked, wearing nooses around their necks, and carrying the keys to the city and castle

One way of seeing it that the "Turnbulls" re-enacted the scene in a hint to that famous episode.

Solution 2:

I think the "will" here means "wish, desire, decision, choice" and so on. It means they were submitting to the King, putting themselves at his disposal, to do with as he pleased. Obviously he was pleased to put a sizable portion of them to death. Probably what confuses the issue is the preposition in, which we would not use today in conjunction with will. We would probably say "at the King's will" now.