Implied subjects for the second person are normal; most sentences in Hebrew (as in English) that address someone as "you" don't name who is being addressed. So an implied "O my soul" is technically possible. (An implied "soul" is also the traditional explanation in Jewish commentaries for the feminine verb in 2 Samuel 13:39, but I prefer to analyze it as masculine; see this article for the theoretical basis.) However, I think it's very unusual to address your soul without naming it. Usually, in the Psalms, the soul is addressed by name (103:1, 104:1, 146:1), which avoids the problem of having to rely on an implied subject when speaking to yourself in second person.
At the level of the consonantal text, I think the best way to understand the word is to analyze it as first person, i.e. as אָמַרְתִּי "I said" without an implied "my soul." This is the same spelling used for the first-person verb form יָדַעְתִּי in the ketiv of Psalms 140:13 (ידעת). The fact that the qere changes the spelling there seems to indicate that ידעת is a valid orthography for a first-person verb (it's also the ordinary orthography in the oldest inscriptions, which don't mark most final vowels). Psalms 140:7 also provides a good parallel (אָמַרְתִּי לַיהוָה אֵלִי אָתָּה) to 16:2 (אָמַרְתְּ לַיהוָה אֲדֹנָי אָתָּה) which also supports reading 16:2 as first person.