You're gonna need damaging bombs for that, of course. That makes your selection somewhat small. You can use:
- Nitronome/Irontech Destroyer (if you're on a budget, consider Irontech Bomb, the 4* variant. It's quite decent at level 10)
- Dark Briar Barrage
- Dark Retribution
- Electron Vortex+Voltaic Tempest
- Deadly Crystal Bomb
- Ash of Agni
And that's about it. Bombs are, in my opinion, largely a support weapon type in SK. Quite a bit like tanks in other games, really. You can expect lots of enemies attacking you because of the aggro generated by the bombs and you'll have to manage that, but that's what bombs are good at.
Relying on area damage is both for better and worse as a solo player. Killing specific enemies, such as Menders and Silkwings can be a pain since there's no way for you to target those specifically but on the other hand, bombs are often the best at taking care of large groups of enemies. For example, a Dark Briar Barrage is by all means the most effective way of clearing a wolver den.
All in all I'd say it's more effective than pure swords and about as effective as pure guns (maybe less), but the best way of playing solo is to have a large arsenal so that you can adapt well to any situation. Spiral Knights is in my opinion not a game that fits playing solo well, so it's often quite hard. Especially if you have a narrow weapon selection. Teammates and complement each other, but as a solo player you have to be several players in one at times.
Should work reasonably well if you have Dark Retribution and Dark Briar Barrage. You could use Nitronome too if you don't mind the flashing. Or you could play it safe and use Shivermist as one of your bombs.