Learn English – Idiom/single word request for ignoring something wrong because it benefits one

To "rob someone blind" means to rob that person as though he or she were blind and thus couldn't detect the robbery. The implication, as medica says in her answer, is that the robbery is likely to be thorough and devastating because the robber has no fear of detection and no need to act in haste. The usage originated in the United States and goes back to the 1890s, though it seems not to have caught on in published writings until the first decade of the twentieth century.

Here is the entry for the idiom in Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013):

rob someone blind Cheat someone in an unusually deceitful or thorough fashion, as in The nurse was robbing the old couple blind. This idiom may allude to robbing a blind beggar, who cannot see that the cup collecting donations is being emptied. {Mid-1900s}

The earliest instances of "robbed blind" that my Ngram searches of Google Books found is in a letter from Milo, of Des Moines, Iowa, published in The Western Druggist (October 1891):

So many druggists have bought cash registers, and are using them with such great satisfaction, that the subject merits at least a passing notice. An inquiry among possessors of these machines, reveals the fact that the motives which induced them to purchase were somewhat mixed. By some they are regarded as a convenience only in making up cash. Others argue that they teach proprietors and clerks to be more exact and systematic. But by far the greater number rely upon them as a partial check upon dishonest clerks, and undoubtedly they are, though you must emphasize the word "partial." The machine which will catch a thief every time, and for sure, has yet to be invented. It's no great trick to beat the registers if one is so disposed. Probably no system of book-keeping in vogue, is any more complete and secure than bank methods, but how often banks are robbed blind by their cashiers? Canada wouldn't have so many inhabitants, if machinery could successfully cope with human ingenuity.

Two other nineteenth-century instances of this wording occur. In a submission by "the Indicator's Own Correspondent," "Binghamton, N. Y., Correspondence," in The Indicator ("A National Journal of Insurance" published in Detroit) (May 1893):

None of the insurance men here have reason to complain because there is not plenty of business. As a matter of fact, there is too much business. They are kept busy now settling fire losses. ... At the O'Neil block fire, which occurred last month, some egregious errors were made, and it is claimed that Mott Boss, who conducted a first-class saloon on the first floor, was robbed blind. The fire broke out at an early hour, and the nearest alarm box, some twenty rods away, refused to respond.

And from "Timber on the Chippewa Indian Reservations," in Congressional Serial Set (1898) [snippet]:

But any change not to take effect until next year should eliminate the Indian logger, because (1) where the Indian logger has sense enough to attend to his own business he is a half-breed, or wife of a squaw man, and his or her success is cause of trouble and jealousy among the tribe; (2) where he is a full blood he must either have a Government man run his whole business or he will be robbed blind, and this robbery will also cause trouble; (3) every Indian wants some one to make a logger out of him, and there are not chances for all of them; the disappointed ones make trouble

The first Google Books instance of "rob [someone] blind" is from a pieced-together article called "Utah's 'Industrial' Army," with no named author, in The [Salt Lake City] Deseret Weekly (May 19, 1894):

A rather singular and ridiculous coincidence is related concerning a resident of Provo who declaimed loudly against the authorities for stopping the "army." They were, he said, free, independent, American citizens, and should be allowed to proceed on their journey uninterrupted. Later in the evening the Carterites were seen prowling about the gentleman's ware house in a suspicious manner, and he at once asked that official protection be given to his property. He was told in a rather curt manner that the men were "free, independent, American citizens, and should be permitted to proceed uninterrupted." He saw the condition in which he had placed himself and changed the tone of his request as follows: "Well, then I want somebody to see to it that this — mob does not rob me blind tonight."

The next instance that my searches turned up is from 14 years later, in "Report of Sixth Vice President Hannon," in Machinists' Monthly Journal (July 1908):

It is about time that some labor leaders, as well as the rank and file, recognize their own weakness. Too often they become the victims of some smooth, oily-tongued official who gives something with one hand in order to rob them blind with the other hand. The sooner the rank and file see the necessity of increasing the dues to $1.50 or $2.00 per month and putting $1.00 or $1.25 a month in the Grand Lodge treasury and amend the laws so that their money can be used for the purpose of financing a strike properly, the sooner we will be able to win strikes quickly or else avoid them entirely.

Other instances follow soon after. E.A.H. Tays, "Mining in Mexico, Past and Present," in Engineering and Mining Journal, volume 86 (October 3, 1908):

The native miner is a curious study. He is hospitable and improvident, superstitious and reckless; is in fact a fatalist. He will work hard, and when treated properly, responds by a frank adherence to his patron, though in the mine he will rob him blind: but then, finds of that kind are God-given, according to his reasoning, and that is not stealing.

Don A. Frantz, The Silent Partner, volume 4 (1909) [snippet]:

The very men who so generously allowed a dying hero three dollars a week, put through scores of pension claims for friends and for political effect, not one of which was untainted of fraud.

That's the kind of pusillanimous gents we pay $5,000 a year to run our affairs and rob us blind.

William Frederick Kirk, "The $11,000 Beauty," in Right Off the Bat: Baseball Ballads (1911):

Rich merchants criticised McGraw in terms that were unkind —

Merchants with lazy shipping clerks and men that robbed them blind.

But Mac just smiled and held his peace. He should have said: "Don't whine!

Mismanage your own business, boys, and let me manage mine!"

From Fur-fish-game, volume 16 (1912) [snippet]:

The fur catch remains only fairly good, as regards quantity, but some of the individual shippers, deluded by high quoting price lists, have had rather bitter experiences; some of these high quoters, by paying low, have made a "pot of money" fooling shippers in the states, but we have been told that when it comes to Alaska shippers, they "rob them blind."

George H. Dacy, Fauquier County, Virginia, letter to the editor, in The Southern Planter (October 1913):

Virginia farmers who are propitiously located as regards railroad facilities and accessibility to a central market should practice winter dairying. Some dealers will buy cream on the butterfat basis irrespective of its condition. It does not have to be produced in an inspected barn. As regards the campaign for sanitary dairying such buyers are to be condemned. From the standpoint of the farmer of small means who wishes to earn the wherewithal to build a better barn they spell profitable possibility. It is essential that the farmer who sells butter fat should maintain a small Babcock testing outfit so that he can run a check on the results obtained by the buyer as otherwise unscrupulous dealers are prone to rob him blind.

The most famous author among earlier users of the phrase is Ring Lardner, in his story "The Crook," in Saturday Evening Post (June 24, 1916):

They was a letter from the girl waitin' for Bull that evenin'. She'd heard from her brother and she knowed that he wasn't burnin' up the League; but he'd confessed that Connie hadn't treated him good and the umpires had robbed him blind. She knew, she wrote, that Bull wouldn't cheat him; if Bull really cared for her, he'd help him if he got a chance. And it would kill her and her father and mother besides if Martin had to face the disgrace o' not makin' good.

In the labor unrest of the late 1910s and 1920s in the United States, the notion of being robbed blind became a popular figure of speech in pro-union periodicals. In just the years 1918 through 1920, Google Books finds versions of the phrase from The American Flint ("Official Magazine of the American Flint Glass Workers' Union of North America"), The Railroad Worker, Utah Labor News (reprinted in National Labor Digest), Brotherhood of Locomotive Firemen and Enginemen's Magazine, and a book-length critique of "the crimes perpetrated by big business" titled Who Will Answer for Mr. Schwab? (invoking the name of Charles M. Schwab, then president of Bethlehem Steel Corporation).

Onelook suggested frazzled, which I think is rather fitting.