There are two verbs formed from the same root, with approximately the same meaning:
1. ἱκνέομαι, a middle (medio-passive) deponent, with a second
aorist, ἱκόμην, inf. ἴκεσθαι. This is the verb in Il. 1.19.http://www.perseus.tufts.edu/hopper/morph?l=i%28%2Fkw&la=greek&can=i%28%2Fkw0&prior=i(kne/omai#lexicon
2. ἵκω, active, also with a second
aorist, but a sigmatic one, ἷξον. (Usually sigmatic aorists have the -α endings.)http://www.perseus.tufts.edu/hopper/morph?l=i%28%2Fkw&la=greek&can=i%28%2Fkw0&prior=i(kne/omai#lexicon
This is confusing, but if you're reading Homer for the first time, you need to become accustomed to the proliferation of alternative forms. Having several forms for the same grammatical category of the same verbal concept made it convenient for someone composing oral poetry in performance to fit an appropriate form in a particular metrical slot. Whether or not the texts of the Homeric poems were actually composed in performance, it's clear that they were the product of a tradition of oral performances.
Also, note that ἱκέσθαι is parallel to ἐκπέρσαι, which is also aorist.