Broken swords were recycled. Depending on how it broke, and where on the blade, it could be made into a shorter sword or dagger/knife. If the tip broke off, a new tip could be ground onto the broken end (and would be lack the hardened edge, which is how this kind of modification can be recognised). If it broke further down the blade, the broken tip could be turned into a dagger/knife (tanto) if long enough.

Otherwise, the steel could be recycled. One application where recycled sword blades were a recommended material was steel-faced armour (where lamellae or helmet plates were made with iron and steel plates forge-welded together). Iron and steel were valuable enough to be traded and sold, so a broken sword had significant scrap value.

As for the samurai who needed a new sword: he bought a new sword.

I know that Samurai viewed their sword as an extension of their own soul.

This idea comes from the quite post-samurai book, Inazo Nitobe, Bushido: The Soul of Japan, originally published in 1905. This book is propaganda aimed at the West, and presents an idealised romatic picture of the samurai and bushido.

To samurai, swords were weapons and badges of rank. Strictly, the badge of rank of samurai was the pair of swords - the daisho, "large and small", consisting of the daito ("long sword", or katana) and shoto ("short sword", or wakizashi). Samurai were sensitive to status and rank, and the right to wear the daisho was important. Nonetheless, they would sell and buy swords (e.g., to upgrade to a better one), pawn them if in need, break them accidentally, give them as gifts, etc.

Basically, Nitobe's idea of the sword as the "soul of the samurai" (as his relevant chapter heading says) is fiction rather than reality.

For some further reading on bushido and Nitobe, see

• Rich, "Bushido: Way of Total Bullshit", 2014: https://www.tofugu.com/japan/bushido/