Hi,
Capacitors are used for many reasons; in a power stage they are generally used to keep the voltage stable by providing a charge reservoir for quick current delivery. Keeping the voltages stable helps reducing the EMV emissions.
In your case the capacitor seems (be careful: we are only looking at pictures, so no diagnosis can be 100% certain) to be just connected in parallel with the power input.
Before doing anything else, you should check if the capacitor is now short-circuited (just measure the resistance in both ways using a multimeter). Then remove it and check if the short is localized in the condensator or further in the circuit.
Then, you can try supplying the laptop with the correct 5V jack, or even better you can provide the 5V using a current-limited power supply. Start very low, like 100mA or less, and work your way up to the maximum current, i.e. until the current stops increasing upon increasing the current limit.
In order to set a reasonable current threshold, look at the power supply. How much current should it provide? Let's say 5V 2A. You set the power supply at 5V and 2,20A max.; if the current keeps increasing, there is someting wrong.
If, instead, everything works fine, well… the capacitor is the only victim, you have been very lucky! :-)
Once you can confirm the capacitor is the only broken part, we can look for an adequate substitute.
Do you feel confident desoldering such a capacitor?