It’s hard to find exactly where “the King’s Gate” was. Pepys refers to the New Cockpit as being “by the King’s Gate”. A sessions book from 1700 says:

Order to remove the watch-house placed near the King’s Gate in Grays Inn Lane, in the parish of St. Andrew’s, Holborn, upon the petition of the inhabitants of the said Lane, that it stands very inconveniently and is of no use or safety to them. The said inhabitants are to erect the watch-house in any other place in or near the lane, as shall be thought fit and safe.

Daniel Defoe also mentions it:

From hence we go on still west, and beginning at Gray’s-Inn, and going on to those formerly called Red Lyon Fields, and Lamb’s Conduit Fields, we see there a prodigious pile of buildings; it begins at Gray’s-Inn Wall towards Red-Lyon Street, from whence, in a strait line, ‘tis built quite to Lamb’s Conduit Fields, north, including a great range of buildings yet unfinish’d, reaching to Bedford Row and the Cockpit, east, and including Red Lyon Square, Ormond Street, and the great new square at the west end of it, and all the streets between that square and King’s Gate in Holbourn, where it goes out; this pile of buildings is very great, the houses so magnificent and large, that abundance of persons of quality, and some of the nobility are found among them…

On the map I have positioned the gate on Gray’s Inn Road, as close to the position of modern-day Cockpit Yard (and therefore Pepys’ New Cockpit) as possible.