What place should prayer as in the Liturgy of Hours (limited because of work) have in relation to Lectio Divina?
On the one hand, for me (but not for many other authors/theologians) Lectio Divina is part of the Mass, or better said: it proceeds from it. In fact, Lectio divina is about taking time in order to receive/digest the Table of the Word of God of today's Mass. So, for me, your question is similar to this one: "What place should prayer, as in the Liturgy of Hours, have in relation to the Mass?"
On the other hand, the Liturgy of the Hours is normally composed of texts taken from Scripture: Psalms, a short reading in all the offices except the "Office of Reading", which consists of a much longer reading. Certainly, in addition to the first "use" of the Divine Office (see below), while we pray the Liturgy of the Hours, Jesus feeds us, and often uses a word or a verse to speak to us. This is really beautiful. Obviously, we are invited to put into practice what He gives us at that moment. So it looks like a short (like an arrow) lectio divina. As you see, reading/praying the Psalms is also a time when God speaks to us. We use His Words to speak to Him (the Psalms) and He uses His Words to speak to us.
The spiritual Power and the ease God puts into speaking to us while we are praying is very interesting, lectio divina wise (PS Please see John of the Cross, "Ascent of Mount Carmel", book II, chapter 26 where John of the Cross explains the power of such words that God give us from time to time.). This is why, while praying the Psalms, it is important to have an "abandoned" attitude (entrusting, surrendering, passive listening) while praying, and not forcing, or paying such excessive attention to the singing (some communities do sing the Psalms, as is normal) to the point of losing the right inner attitude. So, during the Divine Office, we should adopt the same attitude we have while doing the Lectio divina - which is an attitude we have to learn, foster and allow to grow in quality in us. It is important to avoid the attitude of "possessing" the Sacred Text or our prayer (God is the one who prays in us). Praying requires a lot of humility, and praying always starts from God, comes in us, and through us and with us and goes back to God. All this is done by the Holy Spirit.
As you see there is a huge similarity between this aspect of the Divine Office and the lectio, but still, these are two totally different exercises or types of prayer. The Divine office is absolutely important in order to carry our Priestly duty (the priesthood of the faithful), carrying our day, ourselves, all the persons we met, carrying them and entrusting them to the Hands of God. The Divine Office is the Prayer of the Church that intercedes for the World and our participation in it increases the power of transformation God puts in the world. The Divine Office is the Prayer of Jesus who intercedes for us, it is our way to say to God: "all what I do during this day means nothing if your Hand is not in it".
I see here another question: Will what we receive in the Divine Office be different from the light of the day received in the Lectio divina? Remember, my perception of how we spend the day after having done the Lectio divina: we spend it under the one Light received in the morning in the Lectio divina. Do you remember as well how I explained that the food of the day is quite unified, since it is about us (and our need) and Jesus (one line (light) can be drawn between two points (Jesus-Me)). I would tend to believe that the lights we could receive during the Divine Office are of the same kind of light, fostering, echoing the light received in the Lectio Divina. Remember, as I said in my presentation of the lectio divina, there are echoes during the day, echoes of the Light received in the Lectio. So, you may see it in this (unified) way.
PS But the lights we receive during the Divine Office can be more immediately related to our need now and then, in order for us to be freed from the dust that accumulates during work, and be lighter in order to be elevated by God in Prayer, Worship and Praise.
There are various schools of thought in understanding as to how our attitude should be during the Liturgy of the Hours. The span is wide, from one extreme to the other: from the ones who say that we have to be very focused, concentrated, and understand what we read/say/pray/sing, to the ones who push toward a more contemplative attitude (some say: well, having it in latin - as they use to have before - allows us to pray better since we don't understand (funny, but not a stupid statement)), closer to the "Prayer of the heart attitude".
PS Learning the Prayer of the heart helps a lot to have the contemplative way of praying the Divine Office.
Let us not forget as well the importance of a theological education, according to the biblical theology and spirituality of the fathers of the Church on how to understand the Psalms spiritually. Saint Augustine here, in his commentary on the Psalms is a real master. (There is a beautiful booklet in French that will help us discover Saint Augustine with the Psalms: St Augustin "Prier Dieu avec les Psaumes", old book, but has been re-published.)
Now, one last point you mentioned: the limited time to fit in everything.
We only have 24 hours, and, as lay people, we have little time to pray or at least we think so. Trust me, priests (and monks/religious, you'd be amazed) as well would say the same, because everybody has to work and everybody has to manage his time: work vs. prayer. This is why I invited you to do this prayer fervently and with deep conviction: "God, I know how Lectio divina is important, so please, organise my life in a way so that I can put You first ". And then wait and see what He does. A very efficient prayer!
Now, even though, we wouldn't have time to do everything. Discernment is absolutely necessary. Mass is obviously the highest point in our day, if we can attend it. But, it is as important to digest the substantial food we receive in the Mass. And in order to do so, we need to have the two extensions of the two tables of the Mass: Lectio divina (extension of the word of God) and Prayer of the heart (extension of the Communion). Sometimes, it hurts to think that routine can get into the Mass. Lectio divina is about Conversion, changing, improving. I would prefer sometimes to think of that seriously, otherwise, if you not take care of digestion/assimilation, we might end up by relying on our faithfulness to attend Mass, and forget our faithfulness to the Holy Spirit and Jesus. It is good to do both. Let us remember Jesus' words: true worship is to be done in "Spirit and in Truth".
Discernment pushes us to analyse the strength, the quality of certain types of prayer and to prefer them, or give them the priority.
I wouldn't easily dismiss lectio divina, and Council Vatican II did place a great emphasis on not forgetting the Word of God in our prayer life.