Trump faces two problems that did not confront Jackson. First, in response to the patronage system, Congress instituted regularized procedures for filling most administrative offices. The result is that most bureaucratic functions are immunized from political interference by the president because staff members are hired or fired based on merit. This is not to say that the bureaucracy cannot behave politically — it can, and many agencies are partial to liberal causes — but rather that it is often hard for political officials to influence their behavior directly. Jackson, on the other hand, could clean house because there were hardly any laws that limited presidential discretion on appointments. In the case of the Justice Department, Trump basically has to ask them to investigate themselves, rather than just fire them — as Jackson no doubt would have.
Second, Jackson’s presidency came after more than a quarter-century of one-party rule in the United States. Thomas Jefferson, James Madison, James Monroe, and John Quincy Adams were all (Jeffersonian) Republicans. During this time, the Federalist party of George Washington, John Adams, and Alexander Hamilton atrophied, such that by the middle of the 1820s, many Americans thought that party politics were permanently a thing of the past. Jackson cut through this consensus like a hot knife through butter. And it was up to opponents of the new president to organize themselves into a durable political organization. This took time, and the Whig party did not take full shape until the administration of Martin Van Buren, after Jackson had left office.