RX 550 or GT 1030 are super low budget.
Both don't need an additional power cable and should work even with low watt PSU's. (250 - 300 Watts more than enough)
They will get you around 170 - 70 FPS in League, never lower than 70 FPS.
I'm currently using a Sapphire RX 550 myself, due to the mining prices making many cards unaffordable when I built my new PC.
For League only, the 2 GB GDDR5 RAM versions are more than enough.
Becareful with the GT 1030, as some versions come with GDDR4 RAM, do not get those under any circumstance, that's just nVidia ripping people off again.
The GDDR5 version however is perfectly fine and performs just as well as the RX 550
If you plan on getting a GTX 1050 Ti, make sure it is the Ti version and that it has 4 GB GDDR5 RAM.
It is the better card of all 3, however currently it has the worst price to performance balance of all, just because of how the mining boom f'd up all the card prices.
Get the 1050 Ti only if you plan on playing way more than just League / eSports type of games.
For LoL, Dota 2, Overwatch, Fortnite, etc. both, the RX 550 and GT 1030 are more than enough for at least stable 60 FPS at 1920x1080px and medium to maximum graphics settings.