Always good to see a fellow JQA enthusiast
The basic problem with the Adams presidency is that it was hamstrung from the start by the Jacksonian allegation that his election in 1824-1825 was in some way illegitimate. This meant that relations with Congress were pretty poor from the very start, and got even worse after the 1826 elections.
For Adams to have more success with his programme, I think we need to remove the whole 'corrupt bargain' charge. One option is have him win the 1824 election outright, but that seems pretty tough—there were at least seven candidates vying for the presidency in 1824 and even if you take (say) Jackson or Crawford out, I think that would just create room for somebody else. With so many candidates in the race, I'm not sure there's a way of avoiding the contingent election, so we have to make the best we can of it.
Ideally, Adams needs to finish first in the electoral college; this removes the sting from the Jacksonian allegation that their man is somehow the rightful president. The popular vote is, IMHO, less important—too many states didn't use it as a method and anyway, I'm not sure whether anybody was adding it up nationally in the way we do with hindsight.
So, the easiest way is probably for Adams to win New Jersey over Jackson. If everything else stays the same, this leaves us with:
Adams 92
Jackson 91
Crawford 41
Clay 37
The election goes to the House as OTL and as OTL Adams is elected on the first ballot.
Here is the tricky bit. Adams placing first in the EC will remove
some of the strength of the 'corrupt bargain' charge but if Clay is appointed as secretary of state, it still has a bit of power, since Adams wins in the House by winning states like Kentucky and Missouri that voted for Clay first and Jackson second.
The trouble is, Clay
really wants the secretary of state job—he was cross about being passed over for Adams by Monroe in 1817 and he probably feels with some justification that he's the highest ranking Republican with the necessary experience. If Adams doesn't offer it to him, he might well throw his lot in with the opposition, which definitely dooms the Adams presidency.
So, we need Adams to offer him the job, but ideally for him to be smart enough to decline it. There is some possibility of this; Clay himself remarked later that he should have turned in down in OTL. Let's say he's prescient enough to do so, and somebody like Richard Rush becomes secretary of state, at least for the first year or so.
So where does that leave us? Obviously, the Adams administration is much stronger but also if Clay is still speaker of the house then both Congress and the White House are firmly lined up being the kind of nationalist programme you're envisaging.
TL;DR: Adams wins New Jersey—wins the House election—Clay turns down the secretaryship of state.