The Republican Party flipped the bird to Trump, arguably for good or ill reason, throughout the leadership campaign right to the day before the election. Trump attached himself to the GOP for the sole purpose of being himself, whether the Party of Lincoln liked it or not. In the end, enough Republicans, en masse, chose to ignore the pundits and vote him in. This coincided with traditional Obama supporters not showing up for the vote, en masse, in support of Hillary.
In spite of ongoing post inaugural efforts to weaken Trump's administration, the GOP cannot compete with the attention Trump garners. In fact, their effort has been embarrassing. For instance, it is about to cart out their health plan in a plodding manner while Trump deals with ISIL, taliban, with success- an established record of governance.
And yes, Trump will seek support from the Democrats It is not about Party politics. His kryptonite is loss of public interest.