Question about RS flip flop
BASIC CONCEPT:

This works if you're using gates and you limit the output current from the gates  so their output voltage doesn't fall below the inputs' low threshold voltage. 



YOUR CIRCUIT

Here's your circuit and its circuit list in LTspice so you can simulate it.

S1 and S2 correspond to the form "A" switches you've shown, but LTspice doesn't have manually operated switches so I've shown them as relays.

The flip-flop kinda works, but notice that the LEDs are in parallel with base-emitter junctions, so they'll never see more than about 800 millivolts, tops, across them, which means they'll never light up. 

Probe around and you'll find some other discrepancies...



Version 4
SHEET 1 880 680
WIRE -352 -208 -576 -208
WIRE -144 -208 -352 -208
WIRE 432 -208 -144 -208
WIRE 640 -208 432 -208
WIRE -352 -144 -352 -208
WIRE -144 -144 -144 -208
WIRE 432 -144 432 -208
WIRE 640 -144 640 -208
WIRE -400 -128 -512 -128
WIRE 800 -128 688 -128
WIRE -400 -80 -448 -80
WIRE 736 -80 688 -80
WIRE -352 -16 -352 -64
WIRE 640 -16 640 -64
WIRE -144 0 -144 -64
WIRE 64 0 -144 0
WIRE 432 0 432 -64
WIRE 432 0 224 0
WIRE -144 64 -144 0
WIRE -144 64 -240 64
WIRE -48 64 -144 64
WIRE 432 64 432 0
WIRE 432 64 336 64
WIRE 528 64 432 64
WIRE -352 112 -352 64
WIRE -304 112 -352 112
WIRE 64 112 224 0
WIRE 64 112 16 112
WIRE 224 112 64 0
WIRE 272 112 224 112
WIRE 640 112 640 64
WIRE 640 112 592 112
WIRE -144 160 -240 160
WIRE -48 160 -144 160
WIRE 432 160 336 160
WIRE 528 160 432 160
WIRE 64 176 64 112
WIRE 224 176 224 112
WIRE -576 192 -576 -208
WIRE -448 192 -448 -80
WIRE 736 192 736 -80
WIRE -576 320 -576 272
WIRE -512 320 -512 -128
WIRE -512 320 -576 320
WIRE -448 320 -448 272
WIRE -448 320 -512 320
WIRE -144 320 -144 160
WIRE -144 320 -448 320
WIRE 64 320 64 240
WIRE 64 320 -144 320
WIRE 224 320 224 240
WIRE 224 320 64 320
WIRE 432 320 432 160
WIRE 432 320 224 320
WIRE 736 320 736 272
WIRE 736 320 432 320
WIRE 800 320 800 -128
WIRE 800 320 736 320
WIRE -576 416 -576 320
FLAG -576 416 0
SYMBOL npn -304 64 R0
SYMATTR InstName Q1
SYMATTR Value 2N3904
SYMBOL npn 16 64 M0
SYMATTR InstName Q2
SYMATTR Value 2N3904
SYMBOL npn 272 64 R0
SYMATTR InstName Q3
SYMATTR Value 2N3904
SYMBOL npn 592 64 M0
SYMATTR InstName Q4
SYMATTR Value 2N3904
SYMBOL res -160 -160 R0
SYMATTR InstName R2
SYMATTR Value 100
SYMBOL res 416 -160 R0
SYMATTR InstName R3
SYMATTR Value 100
SYMBOL voltage -576 176 R0
WINDOW 0 7 4 Left 2
WINDOW 3 8 106 Left 2
WINDOW 123 0 0 Left 2
WINDOW 39 0 0 Left 2
SYMATTR InstName V1
SYMATTR Value 5V
SYMBOL LED 48 176 R0
WINDOW 0 -78 33 Left 2
WINDOW 3 -129 63 Left 2
SYMATTR InstName LED1
SYMATTR Value NSSWS108T
SYMBOL res -336 80 R180
WINDOW 0 36 76 Left 2
WINDOW 3 36 40 Left 2
SYMATTR InstName R1
SYMATTR Value 100
SYMBOL LED 240 176 M0
WINDOW 0 -82 33 Left 2
WINDOW 3 -129 63 Left 2
SYMATTR InstName LED2
SYMATTR Value NSSWS108T
SYMBOL voltage -448 176 R0
WINDOW 0 12 6 Left 2
WINDOW 3 24 96 Invisible 2
WINDOW 123 0 0 Left 2
WINDOW 39 0 0 Left 2
SYMATTR InstName V2
SYMATTR Value PULSE(0 5 0 10m 10m 80m .6)
SYMBOL sw -352 -160 R0
SYMATTR InstName S1
SYMBOL res 624 80 M180
WINDOW 0 36 76 Left 2
WINDOW 3 36 40 Left 2
SYMATTR InstName R4
SYMATTR Value 100
SYMBOL voltage 736 176 M0
WINDOW 3 24 96 Invisible 2
WINDOW 123 0 0 Left 2
WINDOW 39 0 0 Left 2
SYMATTR InstName V3
SYMATTR Value PULSE(0 5 .3 10m 10m 80m .6)
SYMBOL sw 640 -160 M0
SYMATTR InstName S2
TEXT -562 354 Left 2 !.tran 2 startup uic
TEXT -560 392 Left 2 !.model SW SW(Ron=.01 Roff=1G Vt=2.5 Vh=0)


A SIMPLE FIX
Here's a latch that works OK with one transistor and a 50mA load per leg.

It's an LTspice schematic and I've included the circuit list so you can play with the circuit if you want to.



    Version 4
SHEET 1 1164 772
WIRE -48 -336 -336 -336
WIRE 992 -336 -48 -336
WIRE 992 -288 992 -336
WIRE -48 -272 -48 -336
WIRE -48 -160 -48 -192
WIRE 992 -160 992 -208
WIRE -48 0 -48 -96
WIRE -16 0 -48 0
WIRE 368 0 -16 0
WIRE 960 0 592 0
WIRE 992 0 992 -96
WIRE 992 0 960 0
WIRE 304 64 -144 64
WIRE 1088 64 640 64
WIRE -48 176 -48 0
WIRE 992 176 992 0
WIRE 64 224 16 224
WIRE 160 224 64 224
WIRE 368 224 592 0
WIRE 368 224 240 224
WIRE 576 224 368 0
WIRE 704 224 576 224
WIRE 880 224 784 224
WIRE 928 224 880 224
WIRE 64 384 64 224
WIRE 160 384 64 384
WIRE 368 384 240 384
WIRE 400 384 368 384
WIRE 592 384 544 384
WIRE 704 384 592 384
WIRE 880 384 880 224
WIRE 880 384 784 384
WIRE -336 448 -336 -336
WIRE 400 528 400 384
WIRE 544 528 544 384
WIRE -144 544 -144 64
WIRE 304 544 304 64
WIRE 304 544 -144 544
WIRE 640 544 640 64
WIRE 1088 544 1088 64
WIRE 1088 544 640 544
WIRE -336 656 -336 528
WIRE -48 656 -48 272
WIRE -48 656 -336 656
WIRE 400 656 400 608
WIRE 400 656 -48 656
WIRE 544 656 544 608
WIRE 544 656 400 656
WIRE 992 656 992 272
WIRE 992 656 544 656
WIRE -336 752 -336 656
FLAG -336 752 0
FLAG 368 384 SET
FLAG 592 384 RESET
FLAG 960 0 Q
FLAG -16 0 Q\
SYMBOL npn 16 176 M0
SYMATTR InstName Q1
SYMATTR Value 2N3904
SYMBOL res -32 -176 R180
WINDOW 0 36 76 Left 2
WINDOW 3 36 40 Left 2
SYMATTR InstName R1
SYMATTR Value 24
SYMBOL npn 928 176 R0
SYMATTR InstName Q2
SYMATTR Value 2N3904
SYMBOL res 976 -192 M180
WINDOW 0 36 76 Left 2
WINDOW 3 36 40 Left 2
SYMATTR InstName R2
SYMATTR Value 25
SYMBOL res 800 208 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R5
SYMATTR Value 1000
SYMBOL res 256 208 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R6
SYMATTR Value 1000
SYMBOL voltage 400 512 R0
WINDOW 3 24 96 Invisible 2
WINDOW 123 0 0 Left 2
WINDOW 39 0 0 Left 2
SYMATTR Value PULSE(0 5 0 100n 100n 1m 2)
SYMATTR InstName V2
SYMBOL res 256 368 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R3
SYMATTR Value 1000
SYMBOL res 800 368 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R4
SYMATTR Value 1000
SYMBOL voltage 544 512 R0
WINDOW 3 24 96 Invisible 2
WINDOW 123 0 0 Left 2
WINDOW 39 0 0 Left 2
SYMATTR Value PULSE(0 5 1 100n 100n 1m 2)
SYMATTR InstName V3
SYMBOL voltage -336 432 R0
WINDOW 3 21 100 Left 2
WINDOW 123 0 0 Left 2
WINDOW 39 0 0 Left 2
SYMATTR Value 5V
SYMATTR InstName V1
SYMBOL LED -64 -160 R0
WINDOW 3 23 76 Left 2
SYMATTR Value NSSWS108T
SYMATTR InstName D1
SYMBOL LED 976 -160 R0
WINDOW 0 -16 -5 Left 2
WINDOW 3 -134 76 Left 2
SYMATTR InstName D2
SYMATTR Value NSSWS108T
TEXT -328 704 Left 2 !.tran 10 uic
TEXT 56 512 Left 3 ;NOR1
TEXT 816 512 Left 3 ;NOR2
TEXT 320 192 Left 3 ;A
TEXT 584 192 Left 3 ;A