Any good power supply has a diode across the output terminals, reverse biased. This does two things: it helps protect the circuit when someone connects a battery backwards (sound familiar?), and it may also conduct in some conditions when the power supply is connected in series with another power supply.
When you connected the battery backwards, this diode conducted a large amount of current. A 10 Ah battery is quite large. A smaller battery might have only caused the diode to fail short, but this one might have caused it to fail open. Then the voltage of the battery would be present, in reverse, across the output terminals.
At the very least, find this diode and replace it (note polarity, the cathode band will be at the + terminal). Other devices may have been damaged. The output transistor may be damaged if the voltage was relatively high (causing VCEMAX or VBEMAX to be exceeded). If that is the case, check the components driving that transistor (maybe another transistor or the output of an opamp). The output will likely have an electrolytic capacitor across it, and since these are polarized it may be damaged by the reverse voltage, but not likely.