I hope you connected positive and negative correctly. Check the voltage at the Attiny then - perhaps the schottky diode is defect. You might also have left a solder bridge somewhere. Check all pins against ther neighbors for shorts.
The voltage at the Attiny should be about 0.3 Volts less than at the cell due to the diode. You might have disconnected a pin of the Attiny when you unsoldered the wires. Or it might be fried but they are pretty tough.
Difficult to say what more could be damaged, perhaps if you post a picture.
I didn't desolderd the wires from the MCU, I keep them soldered in order to continue practicing flashing code. I only disconnect the wires from the Arduino board when connecting to battery.
I didn't notice the diode is in series to MCU, so when connecting to Arduino 5v I connected id directly to pin 8
FUSES
* I use these fuse settings
* Low: 0x75 (4.8MHz CPU without 8x divider, 9.4kHz phase-correct PWM or 18.75kHz fast-PWM)
* High: 0xfd (to enable brownout detection)
After some reading in the Attiny Datasheet and other sources about the fuse, mrheosuper was correct and the problem was in the fuse settings.
In case, someone will have the same problem in the future:
The tutorials on the web for flashing MCU using Arduino include burning bootloader. During the burning of bootloader the fuse is also burned.
In order to use the same fuse settings as toykeeper set the BOD to 1.8v and clock to 4.8 MHz internal (under tools menu at the Arduino IDE).