When a rechargeable fan is spoilt from the inside its very easy to bypass the problem and keep using the fan. All the fan needs to rotate is direct current (DC) which is produced by a battery cell inside the fan. So when the whole charging system of the fan spoils and the fan is not working anymore, just cut out the wire coming from the rotating head and connect it directly to the battery (i.e connect the negative wire or blue wire to the negative or blue side of the battery, Do the same for the positive or red wire). Now the issue that'll arise will be getting the battery to charge cos the battery will die eventually.
Now to charge the battery you can use the fan charger (IF ONLY IT HAS ITS OWN BUCK BOOST CONVERTER OR POWER PACK AS IT IS CALLED, AND MAKE SURE THAT IT IS NOT FAULTY) by cutting out the mouth that goes to the fan, then bring out the two wires on the inside the red one is positive and the blue is negative by default, but if yours doesn't have color you can determine which is which using a galvanometer/voltmeter. Now do the same connection for the charger as you did with the fan, i.e plugging positive to positive and negative to negative.
NOW IF YOUR FAN CHARGER IS AN ORDINARY WIRE AND DOES NOT HAVE POWER PACK OR IF THE POWER PACK IS FAULTY. you can use laptop charger with a working power pack to charge it on the following conditions.
1. You must find out the voltage rating of the power pack.
2. You must ensure that the voltage rating of the power pack is greater than the voltage of the battery. The voltage rating of the power pack below is 19 volts.
3. You must ensure that the power pack rating is not more than 5 volts higher than the battery voltage (i.e battery voltage
4. If your fan has two batteries make sure the sum of the voltages of the two batteries satisfies conditions 1,2, and 3. And when connecting the batteries for charging connect it in series.
The above conditions MUST be satisfied for the charging to work. If the voltage rating of your battery is too high you could connect a resistor or any DC appliance that'll lead to a voltage drop.