Maybe some other components are also damaged. There may be broken traces too. It's not only a burnt transistor. You may need to check the whole board with a multimeter. Troubleshooting a PCB without the layout is quite difficult. Redesigning the system is sometimes easier. You can consider...
Here is a good tutorial on interfacing DHT11 with Arduino. if you're still facing a problem, you can cross-check your code and circuit with this one: https://www.theengineeringprojects.com/2017/08/dht11-arduino-interfacing.html
For me, the main problem was that there was more theory than practical in our curriculum. Even in the labs, experiments were done in groups. As a female where the majority were males, I was kinda cornered. At that time there were not so many online tutorials. It was not easy to buy electronic...
Martaine2005 is right. You should solder this loose resistor. But there is no guarantee that the board will be alright after that. There might be some other faulty components too.
There are many videos about repairing samsung washing machine. But sounds like you did not find any solution in those. If so, I think it's better to contact samsung after sales service. Perso ally I prefer taking malfunctioning appliances to the company in the first place. Because warranty may...
They have not uploaded any tutorial. They have not made this project open spource either. So, it is not easy to find out the wiring etc. However, for wearable projects in general, Adafruit Flora and Arduino Lilypad are widely used.