It appears from your example that you've been playing the adventure with a group of 3 characters, without tuning down the encounters.
If you do so, faster leveling is expected.
Rationale
D&D 4e's adventures use encounters that have been balanced against a party of 5, which is the expected number of players. If you have a party of 6 or a party of 4 (or, like in this case, a party of 3), you're supposed to add or remove monsters from the encounters so that the XP parcels scale accordingly.
Your 1300 XP budget encounter should be worth 1300/5=260 XP per character and your characters got 1300/3=433 instead. You should have built an encounter worth 260*3=780 XP
I know this almost nullifies the book's utility. Your statblocks are not that useful when you need to remove almost half the monsters from any encounter.
My personal suggestion
Modify the encounters so that your group is comfortable with their difficulty and then deal out less XP than the encounter is worth (e.g. the example encounter still gives people 260 XP each, even if the rules say otherwise).
You can absolutely use their stat blocks directly as allies. The real question you have is how to make the ensuing encounter appropriately challenging with the additional allies. You have a few ways to handle this.
Increase the encounter to make up for the allies
This takes some tinkering, but the way I would start is take the encounter xp of the additional guards, and create an encounter worth the same xp, assigning monsters from that group to handle the guards. This will make the encounter seem larger, but not actually change what the PC's are handling. By the time the additional mobs take out the guards, they will be weakened, or the PCs will have had the opportunity to take out the primary enemies (or both). You can also fudge it by subtracting the guard's encounter xp value from the original encounter to determine how "hard" of an encounter it actually is.
Don't use stats, just tell a story
The better option from my standpoint is to not bother with tracking hp and stats from the guards, and use them for a story point. If you want one of the enemies to seem tougher than the rest, have him catch a guard's blade in his bare hand and kill him in one blow. If the PC's are hurting, have the guards finish off one of the weaker foes, or take a hit for them. You can still roll for the guards, but use their deaths and kills to make a cinematic point. You can probably come up with a lot of uses for the guards in this context without worrying about their stats.
Do a mix of both
Have the guards statted out, but don't have a fixed encounter size. If the guards start "winning" too much, bring in reinforcements from the assassins. If the players are losing badly, have the guards come to the rescue (or at least take out or distract a couple enemies).
Give the guards a different objective
The guards really don't care about the PC's survival, they care about the King. Maybe they are just fighting to get a clear path to get him to safety. Have them leave the encounter as quickly as possible with the King in tow (or maybe some of the guards are in on the assassination attempt).
There are a lot of options, so don't feel constrained by the rules. You are the DM after all, your job is to make an interesting story/game with the rest of the players and adjudicate rules. Sometimes eyeballing it or making things happen because the story needs it is your job.
Best Answer
Understand it is a Rule of Thumb
So, 6 encounters is a rule of thumb, not a hard and fast "you must make level X after 6 encounters." Also, keep in mind that when it comes to balance and leveling some of the advice in the DMG is hard to follow to the letter. I for instance, have never been able to get the "6 to 8 encounters a day" -- so instead I only use "hard" or "deadly" encounters, which I find my players seem to manage too easily most of the time. They aren't minmaxed, they just play to their strengths in the actual fight.
Likewise for your group, changing the way XP is totaled, or ignoring XP for level purposees might make sense. Let talk about this though.
Not all Encounters are Combat
The books don't talk about the proper amount of XP to give to social encounters, trap encounters, puzzle encounter, etc. But the books and game designers do state that XP should be awarded for them. The reason there aren't charts for them is because they are hard to judge without actually being there. So, if you keep battle XP calculated the same, but start giving XP for these other types of encounters, you might reach the suggestions in the DMG.
For Battle Encounters
The battles your players face will have various numbers of foes. Fewer foes mean larger monsters. This means lower or no multiplier, so the split is better for the players. So, throw in a solo "boss" monster here and there. Or a boss monster with "squishy" minions.
Another option is to look at the published adventures, even if you don't plan to run them. For example Lost Mine of Phandelver, it tell you how many monsters will be in each area. It is calculated that if you run the option first quest the average party should gain one level, before you begin the main story arcs.
How they do manage this? They do a few encounters with a lot of goblins, but by the end you are basically in a mini-boss fight. They've done the math for each encounter ahead of time, and know that running that part will grant X XP for this encounter with a part of Y players.
"Would it be ok if I started adding the multiplier to the actual experience gained?"
Yes. The beauty of the game is that the rules are a guide post. If you do start adding the multiplier, it won't break anything -- but it will make your characters level faster, which will change the feel of the game as character progression would be sped up, but some people like that. If you do, just keep in mind to balance encounters so that they are on the harder side.
Another Option: Milestones
I personally don't use XP. I use milestones. It is more subjective, but it feels more natural to me. With XP, a big boss fight might put you a few points away from the next level, and end up leveling on a random goblin -- might feel anti-climatic. Where with milestones, you kill the dragon -- welcome to the next level.
This also allow the DM to control the pace of the leveling. If you need them to be a certain level for a certain encounter, you can get them there before they reach it.
Question in the title: How to get from 1 to 2 in 6 encounters.
Assume party size of 4 players:
Each player has to get 300 XP. 300 X 4 = 1200 XP.
(using Kobld Fight Club for ease) I generate 6 random encounters... Say 1 easy, 2 medium, 2 hard and 1 deadly. I get:
Total 1,755 XP. Well over the needed 1200 XP. In fact enough XP that a party of 5 players could all make level 2.