Ringed City bosses
The Ringed City has three bosses: Midir, Halflight, and Gael. Before fighting Midir, be sure to grab the nearby Chloranthy Ring+3 and equip it. Next, return to Shira's room, exhaust her dialogue, and you would be able to summon Shira to co-op against Midir.
Darkeater Midir
52. Darkeater Midir, Dark Souls III
Darkeater Midir at SL15, Dark Souls III
The fight against Midir should generally be done solo to prevent it from moving around the arena too much. Before entering the boss arena, cast Tears of Denial. Buff your Raw Dragonslayer's Axe+10 with lightning. Always position yourself in front of Midir and look for opportunities to hit its snout. Get in one or two hits and move away. Midir's attack patterns are easier to observe and dodge when fighting it solo. The video below shows an SL15 character fighting Midir solo. The video is rather long (and the character is very low level) so you could better learn Midir's attack patterns. Once Midir is defeated, return to Shira's room and exhaust her dialogue to receive a Titanite Slab.
Halflight, Spear of the Church
53. Halflight, Spear of the Church, Dark Souls III
Halflight is another gank boss. Lapp is available to be summoned to help you during the fight. The video below shows an easy strategy for defeating Halflight. Equip the Storyteller's Staff, run to the location shown in the video, wait for Halflight to spawn, and use the weapon art of the Staff to poison him.
Wrap up Shira's quest line
54. Wrap up Shira's quest line, Dark Souls III
Before fighting Gael, you should first wrap up Shira's questline. Purchase a Lothric Knight Greatsword from the Shrine Handmaid (if you don't already have the weapon) and upgrade it to +10. Run to the location shown in the video below and wait for Shira to invade you. Equip the Thunder Stoneplate Ring and consume a Yellow Bug Pellet to boost your lightning defense. Buff your great sword with lightning and defeat Shira. Now equip the Flame Stoneplate Ring and consume a Red Bug Pellet to boost your fire defense. Use the same weapon to defeat the nearby lone Ringed Knight to receive a Titanite Slab. Finally, return to Shira's room to collect her armour set.
Slave Knight Gael
55. Slave Knight Gael, Dark Souls III
Gael is weak against poison, toxic, and lightning. Before entering the boss arena, cast Tears of Denial. Start the fight by using your Poison Rotten Ghru Dagger+10 to poison Gael. Once poisoned, switch to your Raw Broadsword+10, buff it with lightning, and wait for an opportunity to get in one or two hits. Rinse and repeat for all three phases of the fight. The poison helps to chip down Gael's HP while you wait for a chance to hit him with your Broadsword. (Poison) Throwing Knife also do chip damage.