In the movie, Leopold is supposed to be the creator of the safety elevator and the founder of the Otis Elevator Company, a multibillion dollar company that is, today, the world's largest manufacturer of vertical transportation systems (elevators and escalators). Otis, in this world, is his valet, for whom he named his company, eventually.
At the point where we first encounter Leopold in 1876, he is clearly having financial troubles, and moreover it's clear that he hasn't yet actually invented anything of import.
Leopold: The monarchy is dead. We are relics. That is reality. The new Royals are men of accomplishment. Men like Roebling, with his bridge, Edison with his lamp,... Diesel, Bell, Westinghouse Those men made themselves from nothing!
Uncle: You, on the other hand, were born with everything... and from it, fashioned nothing.
This clearly shows that though Leopold idolized inventors, he hadn't yet joined their rank. However, in the very next scene, Stuart tells us that in the modern times, he is known as a successful inventor.
Stuart: The Duke of Albany. ... I told you about him. He's a brilliant engineer. He patented the counter-weight pulley. He invented the elevator.
Since the time-travel theory of Kate & Leopold is that nothing can be or was achieved that hadn't already been supposed to have happened all along (Kate goes back because she sees pics of herself in the past, which 'prove' that she is actually supposed to be Stuart's ancestress - this was elaborated in more detail in a rather awkward deleted scene)
Thus, it can be clearly inferred that since Leopold never did, in actual history as known to Stuart, married for money, but had always married Kate, and eventually led to the formation of the Otis Elevator Company, the success of the company, and Leopold finally 'fashioning something' himself, was how their financial troubles were solved.