Flashing and constant LED's help Answered

I like the no microcontroller approach :)
Did you know you can actually get flashing LED's? ;)
Just add power and they flash with nothing else added.
Then there is the option to go 12V and to use a simple and dirt cheap LED(strip) controller.
The keep the last used setting if there is a power loss ;)
So all that would be left is a simple timer for those 10 and 45 seconds.
If in doubt this can be done with a 555 IC and a beefy transistor or better mosfet to give power to the LED controller.
But you can also get timers for 12V in the car department ;)
Using indicator relays for the blinking and the circuit from some windscreen wiper switch that has an adjustable interval knob...
Options are almost limitless once you start thinking out of the box ;)

If for nothing more needed, a very small Arduino is plenty! Like a small Arduino nano or pro micro.
With the arduino, you can PWM each of your LED to dim if you want or make it pulse and flash and whatnot... The sky is the limit... or Kessel ;)

Depending on the power of your LEDs, you may need to feed the PWM thru a FET or BJT for more power...

The LEDs will need a resistor to limit the current. If we know what LEDs you are willing to use, we can calculate the resistors...

The 9V will have to go thru a LDO to lower the voltage to 5V für the Arduino and the LEDs.
I suggest a super simple LM7805 to make 5V from your 9V. Maybe a small heatsink is necessary (Again, depending on the LEDs) and a imput + output cap of 1uF is propably plenty.

I think the cheapest microprocesor that could do this is some kind of ATTiny.

https://en.wikipedia.org/wiki/ATtiny_microcontroll...

Regarding an old-school way to do this. I think the way to do that, is with a 555 timer, or similar, to give a square wave with the smallest period you want. Then use a binary ripple counter, as a frequency divider, to give some signals with lower frequencies, with periods that are powers of 2 multiplied by the shortest period.

For example, if the shortest period is: T0 = 0.33 s

T0*2 = 0.66 s
T0*4 = 1.32 s
T0*8 = 2.64 s
T0*16 = 5.28 s
T0*32 = 10.56 s (close to 10 s)
T0*64 = 21.12 s
T0*128 = 42.24 s (close to 45 s)

I think all of these signals will naturally have 0.5 duty cycle; i.e. spending equal amounts of time on and off; i.e Ton = Toff.

Also I am guessing the timing does not have to be precise, if all this thing has to do is, like, look like it is doing something, to human eyes.

1 x LED flashing how fast 1/10 a second or 1 a second, it is important?
555 timer, a counter and a flip plop.
555 timer makes a signal 0.25 a second.
flip flop makes signal 50% on 50% off 0.5 second on 0.5 second off to LED and to a Decade counter.
Decade counter counts to 10 and signals 1 every 5 seconds to a flip flop to LED that flashes on 10 seconds off 10 seconds.
For a signal 45 seconds on 45 seconds off is a little harder but take the signal from the first counter to a divide by 8 counter will get you close.

you could also use flashing or animated LEDS there are lots of different kinds that change color or flicker and have different flash rates, some have a built in resistor so you can plug in and play
https://www.ledsales.com.au/index.php?main_page=index&cPath=148_178&sort=20a&page=1&zenid=hutpjk0p9b7vvlm8gmaajgkia0