Svanhildr's ability increases the effectiveness of swords by giving them extra damage.
This means that each sword die has 5 hits on two sides, 1 shield, and 3 blank faces.
This does not mean that sword dice become superior to other dice, especially since many enemies have only 1-2 hp. Effectively:
Die | vs 1 hp enemies | vs 2 hp enemies | vs 3 hp enemies
Axe | 4 hits on 4 faces | 6 hits on 4 faces | 6 hits on 4 faces
Spear | 3 hits on 3 faces | 4 hits on 3 faces | 4 hits on 3 faces
Sword | 2 hits on 2 faces | 4 hits on 2 faces | 5 hits on 2 faces
For Svanhildr, swords are, overall, equal to but not better than other weapons. She should tend to accumulate swords not as much because she actively chooses them over other weapons, but because other leaders prioritize spears and axes over swords.
When you're using swords as Svanhildr, it's worth noting that swords are highly effective vs 2 hp enemies (since any sword hit kills immediately), but less effective vs 1 hp enemies. Spears are more reliable members in larger groups; for example, against 3 hp enemies, 2 swords + 1 spear is a little better than 3 swords.
Some spaces other than the Swordsmith provide swords. Note that with Svanhildr in the mix, swords are more likely to run out of supply, so check if there are still sword dice left in the box before taking these spaces later in the game.
Jarl's Hut: This provides a sword. This is a relatively small incentive for other leaders, but a strong incentive for Svanhildr. She does not care about going first any more than other leaders (if anything, the fact that she gets near-equal utility out of all dice make her generally less sensitive to turn order), but the secondary reward is much more attractive.
Varangians, Jomsvikings, Folk Warriors: These all provide swords, and are thus considerably less efficient choices for their cost for other leaders.
Raiders: This is the lone military tile that does not provide swords. If this is on the table, take consolation in the fact that the sword supply is less likely to run out.
The Reaction rune combos with Svanhildr's ability (so swords become 7 hits on 3 faces). Also note that the Potential rune benefits swords (3 blank faces) more than other dice (2 blank faces).
An interesting synergy that showed up in the first game I've played just adding the expansion cards to the deck. (Haven't rolled out the actual full expansions yet, just added the extra cards to the regular decks.)
Svanhildr can automatically kill a Mist Walker by sending two swordsmen. The Mist Walker, added in the Dark Mountains, kills all warriors who roll hits and has four toughness, which means that anybody can guarantee a kill with four warriors (but that you stand a reasonable chance of losing all of them).