Stranded wire was cut... Connecting it back together, with solid wire... Is this safe and possible?
That's neither safe nor robust. The solid wire could break after being flexed, and the tape won't reliably hold it together.

Perhaps more importantly, simply twisting the wires together without solder or the pressure of a crimp or screw terminal will not give a low-resistance, gas-tight joint. The resistance will likely increase over time as the conductors oxidize. The power supply will draw more and more current to compensate for the voltage drop over that resistance. \$P=I^2R\$ and that heat can either directly start a fire or melt the insulation leading to a short circuit.

Buy a new cable, they're incredibly common and cheap.