Republicans will raise the government’s borrowing limit without threatening a default, Senate Majority Leader Mitch McConnell guaranteed Sunday, placating the markets but potentially surrendering the party’s leverage as Congress gears up for another debt showdown. The current debt-ceiling holiday, to which Congress agreed a year ago, expires Sunday, starting the clock on debate and sending the Treasury Department scrambling to tap into its “extraordinary measures” to keep from breaching the limit. Budget analysts say Treasury Secretary Jacob Lew has about seven months of spare room, meaning a deal would have to be reached by October or November. Mr. McConnell, Kentucky...