TL;DR It didn't exactly fail as a country - support for annexation was fairly strong in Texas from the beginning, but it grew as fear of a new war with Mexico made it difficult to attract migrants and raise revenue. Annexation was mostly delayed by US political considerations, esp. slavery and tensions with Mexico.

Andrew Jackson delayed recognizing Texan independence until the last day of his presidency, to avoid political fallout in the election of 1836. Van Buren refused to annex Texas because he also didn't want to upset the balance between slave and free states established by the Missouri Compromise, and because he didn't want to risk war with Mexico (though he was interested in smoothing things over by offering to buy the territory from Mexico.)

Harrison died shortly after taking office, but John Tyler, his VP, promised to annex Texas (partly as a gambit to win some support after his expulsion from the Whig party). Tyler's first attempts at annexation were rebuffed by Congress until he threatened to run third-party and divide the Democratic vote. Texas was annexed at the end of his term, and he threw his support behind Polk, who won the election and sealed the deal.