I'm pretty doubtful that Gian Galeazzo was in financial difficulties.
After all he was the lord of the best agricultural lands in western Europe (lands whose productivity had been even increased by the great irrigation canals sponsored by him) and controlled all the major traffic routes from northern Italy to Germanies and Flanders (which is another way to say that most overland traffic to and from Venice was routed through his lands and paid duties to him).
The larger financial expenditures were also well beyond him by 1402 (I'm talking of the bribes to the HRE to be invested with the ducal title, but also the above mentioned irrigation canals and the drainage of marshes). He had also been re-organizing and improving his chancery and the tax collection (as a matter of fact the first serious census of land properties in Lombardy was commissioned by him).
The siege of Florence was not really very long, considering the difficult terrain protecting the city and the fact that he waited for the surrender of Bologna before investing in earnest Florence. In the last few years he had also become lord of Perugia, Assisi, Nocera, Gualdo Tadino, Spoleto: it's practically all of modern Umbria, and it does not convey the impression of a lord in financial straits.
It's quite true that even after the fall of Florence there are still quite a few independent lordships or free cities in Italy (Savoy, Montferrat, Saluzzo, Genoa in the west; Mantua, Padova, Ferrara, Ravenna, Ancona and Montefeltro in the east, just to mention the most important ones). Note that I do not include Venice or Naples: the Serenissima is the golden goose for the duke of Milan, it would be silly to kill or maim it trying to conquer the city; the kingdom of Naples will require a major effort if it is really to be taken and I believe that it will be a matter that will have to wait one or two generations to be settled (but it is quite likely that from his holdings in Umbria GG will still play his little subversion games with the lords of Abruzzo and Puglia). My guess is that a GG surviving the plague of late 1402 will take a few years to consolidate and re-organize his lordships (except maybe a short campaign against Padova and aiding and abetting his Malatesta allies and vassals to settle their long-standing feud with the Montefeltro). The years up to 1410 should be quiet ones: afterwards the dynastic crises in Aragon and France should open a lot of juicy possibilities.